Math 411 — Introduction to Topology

Course Schedule & Homework Assignments

Here is a link back to the course syllabus/policy page.

This schedule is will be changing **very frequently**, please check it at
least every class day, and before starting work on any assignment (in case the
content of the assignment has changed).

If you see the symbol below, it means that class was videoed and you can get a link by e-mailing me. Note that if you know ahead of time that you will miss a class, you should tell me and I will be sure to video that day for you.

*M:**Content:***Notation:**- $x\in S$ means $x$ is an element of the set $S$.
- $\forall$ should be read "for all"
- $\exists$ should be read "there exists"
- $\Rightarrow$ should be read "implies"
- We write $A\subset B$ if $A$ is a subset of $B$.
- Given sets $A\subset B$, $A^c$ should be read "the complement
of $A$" [remember: this is all elements
**not in $A$**,*i.e.,*$A^c=\{b\in B\mid b\notin A\}$. *s.t.*stands for "such that"

**Definition:**Given two sets $A$ and $B$ and a function $f:A\to B$, for any subset $C\subset B$ of the codomain, we define its**inverse image**$f^{-1}(C)=\{a\in A\mid f(a)\in C\}$ to be those elements of the domain $A$ which get sent by $f$ to points in $C$. [Note: the function does not have to be one-to-one — there does not have to be an inverse function at all — for $f^{-1}(C)$ to exist, despite the similarity of notation to the notation for an inverse function!]**Definition:**Given sets $A$ and $B$, the**Cartesian product**is the set of ordered pairs, the first from $A$ and the second from $B$: $A\times B=\{(a,b)\mid a\in A,b\in B\}$**Definition:**Given a set $X$, a**metric**on $X$ is a function $d:X\times X\to\RR$ satisfying the properties:*symmetry:*$\forall x,y\in X,\ d(x,y)=d(y,x)$*non-negativity:*$\forall x,y\in X,\ d(x,y)\ge0$ and $d(x,y)=0$ if and only if $x=y$.*the triangle inequality:*$\forall x,y,z\in X,\ d(x,y)\le d(x,z)+d(z,y)$.

**metric space**.**Definition:**The**standard**[or**Euclidean**]**metric**on $\RR$ [or $\RR^1$] is $d(x,y)=|x-y|$.**Definition:**the**Euclidean metric on $\RR^2$**is defined as $d((x_1,y_1),(x_2,y_2))=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$.**Definition:**the**taxicab metric on $\RR^2$**is defined as $d((x_1,y_1),(x_2,y_2))=|x_1-x_2|+|y_1-y_2|$.**Definition:**the**[open] ball**$B(a,r)$ in a metric space $(X,d)$ is defined as the set $B(a,r)=\{x\in X\mid d(a,x)\lt r\}$.- examples: discussing what open balls look like in the Euclidean and taxicab metric on $\RR^2$.
**Definition:**In a metric space $(X,d)$, a set $O\subset X$ is called**open**if $\forall a\in O,$ $\exists r\gt0$*s.t.*$B(a,r)\subset O$.

A set $C$ is called**closed**if $C^c$ is open.- examples: discussing which sets are open in $\RR^1$ and $\RR^2$ under various metric topologies
**Theorem:**the collection of open sets in a metric space is closed under arbitrary unions and finite intersections.**Definition:**Given a set $X$, a**topology**on $X$ is a collection $\Oo$ of subsets of $X$, called the collection of*the open sets in this topology*, satisfying the properties:- $\emptyset\in\Oo$ and $X\in\Oo$
- if $\Aa\subset\Oo$ is a any collection of open sets, then
$\bigcup_{O\in\Aa} O\in\Oo$
*["arbitrary unions of open sets are open"]* - if $\Ff\subset\Oo$ is a finite collection of open sets, then
$\bigcap_{O\in\Ff} O\in\Oo$
*["finite intersections of open sets are open"]*

**topological space**.**Definition:**In a topological space $(X,\Oo_X)$, a set is called**clopen**if it is both open and closed. (*E.g.,*in any topological space $X$ both $\emptyset$ and $X$ are always clopen.)**Definition:**Given any set $X$, the**trivial topology**on $X$ is the one whose open sets are $\Tt=\{\emptyset, X\}$ while the**discrete topology**on $X$ is the one whose open sets are $\Dd=\{O\subset X\}$ [*i.e.,*in the discrete topology, every set is open].**Definition:**Given topological spaces $(X,\Oo_X)$ and $(Y,\Oo_Y)$ and a function $f:X\to Y$, we say $f$ is a**continuous function**if $\forall O\in\Oo_Y$ $f^{-1}(O)\in\Oo_X$; that is, the inverse image of every open set in $Y$ is open in $X$.

- As soon as possible, please do
**HW0:***Send me e-mail*(to`jonathan.poritz@gmail.com`)*telling me*:- Your name.
- Your e-mail address. (Please give me one that you actually check fairly frequently, since I may use it to contact you during the term.)
- Your year/program/major at CSUP.
- The reason you are taking this course.
- What you intend to do after CSUP, in so far as you have an idea.
- Past math and science classes you've had.
- Your favorite mathematical subject.
- Your favorite mathematical result/theorem/technique/example/problem.
- Anything else you think I should know (disabilities, employment
or other things that take a lot of time,
*etc.*) - [Optional:] The best book you have read recently.

**Things to think about for next class:**- In $\RR^2$, we have called the ball $B(0,1)$ the "open unit ball". How do we know it is open? What has to be checked for that?
- ...Repeat the last T2TA for a general metric space, not just $\RR^2$.
- We have considered what the open balls look like for the
Euclidean and taxicab metrics on $\RR^2$. Do they induce different
topologies?
*I.e.,*is every Euclidean-open set also taxicab-open, and vice-versa? - We now have three topologies on $\RR^1$: the standard (Euclidean), the trivial, and the discrete. For each choice of one of these topologies on both domain and codomain, can you describe what are the continuous functions $f:\RR^1\to\RR^1$? (There are nine possible combinations of topologies.)
- ...Generalizing the previous T2TA: In fact, if $(X,\Oo_X)$ and $(Y,\Oo_Y)$ are topological spaces with some particular topologies we care about, we can also give $X$ and $Y$ their respective trivial topologies $\Tt_X$ and $\Tt_Y$ and their respective discrete topologies $\Dd_X$ and $\Dd_Y$. Again, can you characterize the continuous functions for each of the nine choices of one topology for the domain and one for the codomain?
- Can you think of a topology on $\RR^1$ which we haven't mentioned yet? Get creative, think of a new one!
- ...Continuing the previous T2TA: what about new topologies on $\RR^2$?
- ...Here's an easier "find new topologies" question: can you list all of the topologies on a finite set? Start with a set that has only two elements, keep going....
- Can you think of any non-trivial (
*i.e.,*not just $\emptyset$ and the whole space) clopen sets in any topological space we have considered? Could you build a topological space which had non-trivial clopen sets?

*W:***Reread the definitions above***[If you want another source for those same definitions, you could look at Chapter 1 of Allen Hatcher's Notes on Introductory Point-Set Topology and/or the first few pages of KC Border's Introduction to Point-Set Topology.]**Content:***Definition:**Given a set $X$ and two topologies $\Oo_1$ and $\Oo_2$ on $X$, if $\Oo_1\subset\Oo_2$, then we say that $\Oo_1$ is a**coarser**topology that $\Oo_2$ is**finer**than $\Oo_1$. That is, a*finer*topology has*more*open sets, a*coarser*topology has*fewer*.**Proposition:**Given a set $X$, the identity map $id:X\to X:x\mapsto x$ is a homeomorphism $(X,\Oo_\text{Euclid})\to(X,\Oo_\text{taxicab})$ and $(X,\Oo_\text{taxicab})\to(X,\Oo_\text{Euclid})$.**Definition:**Given topological spaces $(X,\Oo_X)$ and $(Y,\Oo_Y)$ and a function $f:X\to Y$, we say $f$ is a**homeomorphism**if $f$ is 1-1 and onto, and both $f$ and $f^{-1}$ are continuous.**Definition:**Given topological spaces $(X,\Oo_X)$ and $(Y,\Oo_Y)$, we say $(X,\Oo_X)$ and $(Y,\Oo_Y)$ are**homeomorphic**if there exists a homeomorphism between $X$ and $Y$.**Definition:**Given topological spaces $(X,\Oo_X)$ and $(Y,\Oo_Y)$ and a function $f:X\to Y$, we say $f$ is a**local homeomorphism**if for every $x\in X$ there exists $O\in\Oo_X$ such that $x\in O$, $f(O)\in\Oo_Y$, and, as a map from $O$ to $f(O)$, $f$ is a homeomorphism.**Definition:**Given topological spaces $(X,\Oo_X)$ and $(Y,\Oo_Y)$, we say $(X,\Oo_X)$ and $(Y,\Oo_Y)$ are**locally homeomorphic**if there exists a local homeomorphism between $X$ and $Y$.**Definition:**Given a set $X$, the collection of sets $\{O\subset X \mid O^c\ \text{is finite}\}$ is called the**finite complement topology**.**Definition:**Given topological spaces $(X,\Oo_X)$ and $(Y,\Oo_Y)$, where $X\cap Y=\emptyset$, we define the**disjoint union**$X\sqcup Y$ to be the topological space whose points are $X\cup Y$ and whose open sets are those sets $O\subset X\cup Y$ satisfying $O\cap X\in\Oo_X$ and $O\cap Y\in\Oo_Y$ —*i.e.,*the parts of $O$ in $X$ and $Y$ are open in $X$ and $Y$, respectively.

*[If $X\cap Y\neq\emptyset$, first make new copies of $X$ and $Y$ which***are**disjoint, then do the above.]**Definition:**Given a topological space $(X,\Oo_X)$ and a subset $Y\subset X$, the**subspace topology on $Y$**is the set $\Oo_Y=\{O\subset Y \mid \exists U\in\Oo_X\ \text{s.t.}\ O=Y\cap U\}$.**Definition:**Given a set $X$, a collection $\Bb$ of subsets of $X$ is a**base**if it satisfies:- $\forall x\in X\ \exists B\in\Bb$ s.t. $x\in B$
- given $B_1,B_2\in\Bb$, $\forall x\in B_1\cap B_2$ $\exists B_3\in\Bb$ s.t. $x\in B_3\subset B_1\cap B_2$

**Theorem:**If $X$ is a space and $\Bb$ is a base for $X$, there is a unique coarsest topology $\Oo$ on $X$ such that $\Bb\subset\Oo$.**Proposition:**Given topological spaces $(X,\Oo_X)$ and $(Y,\Oo_Y)$, the set $\Bb=\{A\times B \mid A\in\Oo_X, B\in\Oo_Y\}$ is a base for $X\times Y$.**Definition:**Given topological spaces $(X,\Oo_X)$ and $(Y,\Oo_Y)$, the topology on $X\times Y$ whose existence is implied by*Propositions 8*and*9*is called the**product topology**on $X\times Y$, and will be denoted $\Oo_{X\times Y}$.**Proposition:**The product topology on $\RR\times\RR\ [=\ \RR^2]$ is the same as the topology induced by the Euclidean or taxicab metrics.**Definition:**Given a topological space $(X,\Oo_X)$ and any set $S\subset X$, we define- the
**interior**of $S$, written $\operatorname{int}(S)$ [or sometimes $S^\circ$] is the union of all open sets contained in $S$,*i.e.,*$\displaystyle\operatorname{int}(S)=\bigcup_{\substack{O\subset S\\O\in\Oo_X}} O$. - the
**closure**of $S$, written $\overline{S}$ is the intersection of all closed sets containing in $S$,*i.e.,*$\displaystyle\overline{S}=\bigcap_{\substack{S\subset C\\C^c\in\Oo_X}} C$. - the
**boundary**of $S$ is the set $\partial S=\overline{S}\smallsetminus\operatorname{int}(S)\ [=\ \overline{S}\cap{\operatorname{int}(S)}^c = \{x\in\overline{S} \mid x\notin\operatorname{int}(S)\}]$.

- the

(turn in at least three of these next Monday):*HW1*- Given $a,b\in\RR$ with $a\lt b$, prove that the closure of the
open set $(a,b)$ is the set $[a,b]$.
*[There is not much to do here, so do it very explicitly and carefully.]* - Give $\ZZ$ the subspace topology it gets as a subset of $(\RR,\Oo_{Euclid})$. What are its open sets? Explain. (As always!) Do we have another name for this topology?
- So it is pretty clear that two finite sets with a different
number of points cannot be homeomorphic, no matter what topologies
you put on them — there cannot even be a 1-1 and onto function
between them, no less a 1-1 and onto function which is
continuous with continuous inverse. But what about infinite sets?
For example:
- Is the set of natural numbers $\NN=\{1, 2, 3, \dots\}$ homeomorphic to the set $\NN\cup\{0\}$? (Both being given their subspace topologies as subsets of $\RR$.) If so, construct such a homeomorphism. If not, why not?
- Next, is $\NN$ homeomorphic to the set $I=\{1/n \mid n\in\NN\}$ (again, given its subspace topology as a subset of $\RR$)?
- What about $\NN$ and $I\cup\{0\}$ (subspace topology in $\RR$) — are they homeomorphic?

- Let $X$ be the $x$-axis in $\RR^2$. Is $X$ with its subspace topology homeomorphic to $(\RR^1,\Oo_{Euclid})$?
- Interiors, closures, and boundaries depend upon what space you
are working in. Look:
- Let $A=\{(x,y)\in\RR^2 \mid 0\le x\lt 1\ \text{and}\ y=0\}$ — think of this as a subset of $(\RR^2,\Oo_{Euclid})$. Compute $\operatorname{int}(A)$, $\overline{A}$, and $\partial A$.
- That set $A$ looks an awful lot like the set $B=[0,1)\subset\RR$. Looking at $B$ as a subset of $\RR$, compute $\operatorname{int}(B)$, $\overline{B}$, and $\partial A$.

- Given $a,b\in\RR$ with $a\lt b$, prove that the closure of the
open set $(a,b)$ is the set $[a,b]$.

*M:**Content:*- Going over some of the
*HW1*problems. We noticed the general strategy of finding properties of topological spaces which are defined entirely in terms of constructions with open sets (*e.g.,*one such property might be "there is a point in the space such that every open set containing this point has infinitely many other points") ... which properties are*homeomorphism invariant*, in the sense that if they are true in one space, then they must be true in any other space which is homeomorphic to the original space. **Definition:**Given a topological space $(X,\Oo_X)$ and a set $A\subset X$, we say $L\in X$ is a**limit point of $A$**if $\forall O\in\Oo_X$ such that $L\in O$, $\exists a\in A\cap O$ with $a\neq L$. That is, every open set containing $L$ contains a point of $A$ other than $L$.**Proposition:**Given a topological space $(X,\Oo_X)$ and a set $A\subset X$, let $\Ll(a)$ denote the set of limit points of $A$. Then $A\cup\Ll(A)=\overline{A}$.**Theorem:**A function $f:\RR\to\RR$ is continuous in the sense of calculus (the $\epsilon$-$\delta$ definition) if and only if it is continuous as a function from the topological space $(\RR,\Oo_{Euclid})$ to itself (the*inverse images of open sets are open*definition).**Definition:**Let $C$ be the subset of $[0,1]\subset\RR$ formed by repeated removing the open middle thirds of every interval in the previous step. This $C$ is called the**Cantor set**.**Proposition:**The Cantor $C$ set is the set of points in $[0,1]\subset\RR$ which have a base expression without the digit "1". $C$ is closed in $\RR$, contains no intervals, every point of $C$ is a limit point of $C$ (*i.e.,*no point of $C$ is*isolated*).**Definition:**A set $S$ is called**countable**if there exists a function $f:\NN\to S$ which is 1-1 and onto.**Proposition:**The set $\QQ$ of rational numbers is countable.**Example:**Let $f:\NN\to\QQ$ be the function whose existence is guaranteed by the above proposition. The let $$Z=\bigcup_{j=1}^\infty (r(j)-\frac{1}{2^j},r(j)+\frac{1}{2^j})\ \ .$$ Notice $Z$ is a union of open intervals so is open. Also, $Z$ contains all of the rationals (they are the centers of all of those open intervals), $\QQ\subset Z$. The complement of $Z$ is a closed set which contains no intervals (since there are rationals in every interval).**Definition:**Given a topological space $(X,\Oo_X)$, we say- $X$ is
**T0**if whenever $x_1,x_2\in X$ are distinct points of $X$ (so $x_1\neq x_2$), $\exists O\in\Oo_X$ such that either $x_1\in O$ and $x_2\notin O$ or $x_2\in O$ and $x_1\notin O$ (*i.e.,*one of the points is in the open set $O$ and one is not); - $X$ is
**T1**if whenever $x_1,x_2\in X$ are distinct points of $X$ (so $x_1\neq x_2$), $\exists O_1,O_2\in\Oo_X$ such that $x_1\in O_1$ but $x_2\notin O_1$, and $x_2\in O_2$ but $x_1\notin O_2$ (*i.e.,*both of the points have open sets like those asked for in a*T0*-space); - $X$ is
**T2**if whenever $x_1,x_2\in X$ are distinct points of $X$ (so $x_1\neq x_2$), $\exists O_1,O_2\in\Oo_X$ such that $x_1\in O_1$, $x_2\in O_2$, and $O_1\cap O_2=\emptyset$;

*T2*-space is also called**Hausdorff**.

*[There are also definitions of***T3**and**T4**, but we will not use them in this class.]- $X$ is
**Proposition:**If a topological space is*T2*, then it is also*T1*. If it is*T1*, then it is also*T0*.**Proposition:**In a*T1*space, every point is closed.**Proposition:**If $(X,\Oo_X)$ is a Hausdorff topological space and $Y\subset X$ is given the subspace topology, then $Y$ is also Hausdorff.**Theorem:**A topology coming from a metric is Hausdorff.**Example:**Here's a pretty stupid metric on any set $X$: $$d(x,y)=\begin{cases} 0 & \text{if }x=y \\ 1 & \text{if }x\neq y \end{cases}$$ The metric topology on $X$ induced by this metric is the discrete topology.

- Going over some of the
(due Wednesday):*HW2*- Prove or disprove: $\RR$ with the finite complement topology is
Hausdorff.

*Extra credit:*For which sets $X$ is $X$ with the finite complement topology Hausdorff and for which not Hausdorff? Prove it!

- Prove or disprove: $\RR$ with the finite complement topology is
Hausdorff.

*W:**Content:***Definition:**Given a topological space $(X,\Oo_X)$ and a point $x\in X$, a set $N\subset X$ satisfying $x\in O\subseteq N$ for some open set $O$ is called a**neighborhood of $x$**.*[So, a neighborhood is a set containing $x$ which is at least as big as some open set containing $x$.]*- Before we get started, a quick comment: many constructions in
topology consider a set of objects which contains certain starting
objects and is then closed under certain operations —
*e.g.,*the topology on $\RR$ contains all open intervals and is closed under finite intersections and arbitrary unions. It is important*not*to conclude then that you take the starting objects, do each of the named operations in turn, then you end up with all objects in the collection!*E.g., it is not true*that all open sets in $\RR$ are arbitrary unions of finite intersections of open intervals — maybe an open set is a finite intersection of an infinite union of a finite intersection of open intervals, or an infinite union of such sets, or .... - Recall

Here are some details about how to "make new copies of $X$ and $Y$ which are disjoint":**Definition:**Given topological spaces $(X,\Oo_X)$ and $(Y,\Oo_Y)$, where $X\cap Y=\emptyset$, we define the**disjoint union**$X\sqcup Y$ to be the topological space whose points are $X\cup Y$ and whose open sets are those sets $O\subset X\cup Y$ satisfying $O\cap X\in\Oo_X$ and $O\cap Y\in\Oo_Y$ —*i.e.,*the parts of $O$ in $X$ and $Y$ are open in $X$ and $Y$, respectively.

*[If $X\cap Y\neq\emptyset$, first make new copies of $X$ and $Y$ which***are**disjoint, then do the above.]- Look at $X\times\{0,1\}$ — this is the set of pairs of the form $(x,b)$, where $x\in X$ and $b=0$ or $1$; basically, it is two copies of $X$, one with a "0" next to it, the other with a $1$ next to it.
- Define $X_0=X\times\{0\}\subset X\times\{0,1\}$, being one of those copies of $X$.
- Similarly, define $Y_1=Y\times\{1\}\subset Y\times\{0,1\}$, being a copy of $Y$.
- But now notice that even if $X\cap Y\neq\emptyset$, it is nevertheless the case that $X_0\cap Y_1=\emptyset$ since every element of $X_0$ has $0$ as its second component, while every element of $Y_1$ has $1$ — not the same!
- Now define the function $\pi^X_1:X\times\{0,1\}\to X$ by
$\pi^X_1((x,b))=x$ — this is the
*projection*onto the first component. - Similarly, define $\pi^Y_1:Y\times\{0,1\}\to Y$ by $\pi^Y_1((y,b))=y$.
- We can now define the disjoint union $X\sqcup Y$ to be the topological space with points $X_0\cup Y_1$ and open sets being the subsets $O\subset X_0\cup Y_1$ satisfying $\pi^X_1(O\cap X_0)\in\Oo_X$ and $\pi^Y_1(O\cap Y_1)\in\Oo_Y$.
- This process of "making them disjoint" is technical but
important:
*e.g.,*if $X$ is the $x$-axis in $\RR^2$ and $Y$ is the $y$-axis, then $X\cup Y$ is a subset of $\RR^2$ and so has its subspace topology. This topological space*is not the same*(not homeomorphic to) the disjoint union $X\sqcup Y$! In fact, $X\sqcup Y$ is homeomorphic to two distinct copies of $\RR^1$, such as, for example, the lines $y=17$ and $y=-17$ in $\RR^2$. - A general thing to notice about the disjoint union $X\sqcup Y$ of (non-empty) topological spaces: it has some interesting clopen sets: $\emptyset$, $X$, $Y$, and $X\sqcup Y$ are all clopen in $X\sqcup Y$.

**Definition:**A topological space $X$ is called**connected**if there do not exist subsets $A,B\subset X$ satisfying- $A$ and $B$ are both open
- $A$ and $B$ are both non-empty
- $X=A\cup B$

*Note*that if we had such $A$ and $B$, it would follow that $A=B^c$ and $A=B^c$, meaning that $A$ and $B$ would also be closed. Therefore, this definition is equivalent to saying that $X$ has no clopen subsets other than $\emptyset$ and $X$ itself.

*Note also*that if we had such $A$ and $B$, it would follow that $X=A\sqcup B$. Therefore, this definition is also equivalent to saying that $X$ cannot be written as a non-trivial disjoint union.**Definition:**A topological space $X$ is called**disconnected**if it is not connected.**Definition:**Given a topological space $X$, a subset $S\subset X$ is called**connected**if it is a connected topological space when given the subspace topology.**Examples:**Consider $\RR$ with each of the four topologies we have given it so far in this class:- with the trivial topology, it is connected
- with the discrete topology, it is disconnected
- with the finite complement topology, it is connected
- with the usual (Euclidean) topology, it is connected.

**Examples:**$\ZZ$ and $\QQ$, as subsets of $\RR$ with its Euclidean topology, are both disconnected. In fact, the connected subsets of $(\RR,\Oo_{Euclid})$ are all intervals (open, closed, half-open, infinite, half-infinite, ...).**Definition:**Given a set $X$ and two subsets $A,B\subset X$, the**set difference**is $A\smallsetminus B=A\cap B^c=\{x\in A \mid x\notin B\}$.**Example:**For any $a\in\RR$, the set $\RR\smallsetminus\{a\}$ with the subspace topology coming from $(\RR,\Oo_{Euclid})$ is disconnected.**Theorem:**Let $f$ be a continuous function from a connected topological space $X$ to a topological space $Y$. If $f$ is onto then $Y$ is connected.**Corollary:**Let $X$ and $Y$ be homeomorphic topological spaces. Then $X$ is connected if and only if $Y$ is connected.**Corollary:**Let $f$ be a continuous function from a connected topological space $X$ to a topological space $Y$. Then $f(X)$ is a connected subspace of $Y$.*[This result is usually stated as "the continuous image of a connected set is connected."]***Theorem:**Let $X$ be a topological space and assume $X=A\cup B$ for two non-empty, disjoint, open sets $A,B\subset X$. Say $Y\subset X$ is a connected set in $X$. Then $Y\subset A$ or $Y\subset B$.*[This theorem is strangely reminiscent of***Euclid's Lemma**in number theory.]**Theorem:**Let $X$ be a topological space and say $\{C_\alpha \mid \alpha\in A\}$ is a collection of connected subsets of $X$. Assume $\exists x\in\bigcap_{\alpha\in A} C_\alpha$. Then $\bigcup_{\alpha\in A} C_\alpha$ is connected.*[I.e., the union of a collection of connected sets all of which have the same point in common is itself connected.]***Theorem:**Let $X$ and $Y$ be connected topological spaces. Then the Cartesian product $X\times Y$ with its product topology is connected as well.**Proposition:**Let $A$ be a connected subset of a topological space $X$. Then the closure $\overline{A}$ is also connected.**Definition:**Given a topological space $X$, a continuous function $\alpha$ from the unit interval $[0,1]$ (with its topology as a subspace of $(\RR,\Oo_{Euclid})$) is called a**path from $\alpha(0)$ to $\alpha(1)$**.**Proposition:**Let $\alpha$ be a path in the topological space $X$. Then the range of $\alpha$, being the set $\alpha([0,1])\subset X$, is a connected subset of $X$.**Definition:**A topological space $X$ is called**path-connected**if $\forall x,y\in X$ there is a path in $X$ from $x$ to $y$.**Theorem:**Let $X$ be a topological space. If $X$ is path-connected, then it is connected.**Definition:**If $P$ is a property topological spaces may have, then we use the term**locally $P$**in two ways:- Sometimes we say a the topological space $X$ is
**locally $P$**if $\forall x\in X$, there exists a neighborhood of $x$ which has the property $P$. - The other usage is when we call a topological space $X$
**locally $P$**if $\forall x\in X$ and for all open neighborhoods $O$ of $x$, there exists an open neighborhood $U$ of $x$ such that $U\subset O$ and $U$ has property $P$. We shall use this version, for example, for the terms**locally connected**and**locally path-connected**

- Sometimes we say a the topological space $X$ is
**Example:**Define the sets $S=\{(x,\sin(1/x)) \mid 0\lt x\le1\}\subset\RR^2$ and $I=\{(0,y) \mid -1\le y\le 1\}\subset\RR^2$. Note that every point of $I$ is a limit point of $S$ and in fact $\overline{S}=S\cup I$. It is a fact that $\overline{S}$ is connected but not path-connected.

(due next class, which is next Wednesday):*HW3*- Let $\Oo$ be the set of open sets in the finite complement
topology on $\RR$. We said in class that $(\RR,\Oo)$ is connected.
What about the following sets, each being given its subspace
topology in $(\RR,\Oo)$?
- $\RR\smallsetminus\{0\}$
- $\RR\smallsetminus[0,1]$
- $\ZZ$
- $[0,1]\cup[2,3]$
- $[0,1]\cup\{17\}$

- Let $X$ and $Y$ be topological spaces and $y\in Y$. Prove that $X$ is homeomorphic to $X\times\{y\}$.
- Let $X$, $Y$, and $Z$ be topological spaces and $f:X\to Y$ and $g:Y\to Z$ be continuous functions. Prove that $g\circ f:X\to Z$ is continuous.
- Prove a proposition like the theorem above which tells us that the continuous image of a connected set is connected, but now for path-connected sets. That is, prove: Suppose $f:X\to Y$ is a continuous function from a path-connected topological space to another topological space. Then if $f$ is onto, $Y$ must also be path-connected.
- Prove that $\ZZ$ is locally connected but disconnected, and that
$\QQ$ is both disconnected and locally disconnected.
*[We did discuss these in class, but this is an opportunity to practice writing out a clear, careful, complete proof.]* **Definition:**- Can you think of a space which is locally homeomorphic, but not homeomorphic, to $\RR$?

- Let $\Oo$ be the set of open sets in the finite complement
topology on $\RR$. We said in class that $(\RR,\Oo)$ is connected.
What about the following sets, each being given its subspace
topology in $(\RR,\Oo)$?

*M:***Memorial Day**, no class.

*W:***Read**, if you want another version of the work we are doing in class, with some more detail in some of the proofs, Chapters 2 and 3 of Allen Hatcher's Notes on Introductory Point-Set Topology.*Content:***Notation:**- $f:A\hookrightarrow B$ means $f$ is a 1-1 function from set $A$ to set $B$
- $f:A\twoheadrightarrow B$ means $f$ is an onto function from $A$ to $B$

**Definition:**The**$n$-sphere**is the set $S^n=\{x\in\RR^{n+1} \mid |x|=1\}$.*[Note: the $n$-sphere is***intrinsically**$n$-dimensional — think of the one angular coordinate specifying a point on $S^1$ or the two coordinates (latitude and longitude) which describe points on a globe (being $S^2$) — but it sits inside $(n+1)$-dimensional space.]

A special case of this is the**circle**, another name for $S^1$.- Some discussion of problem 6 from
*HW3*. In particular, we discussed several examples, such as $S^1$ and $\RR\sqcup\RR$ [${}\cong\RR\times\{0,1\}$], of topological spaces which are locally homeomorphic, but not homeomorphic, to $\RR$. - The
**Intermediate Value Theorem:**A continuous function $f:[a,b]\to\RR$ takes on all values between $f(a)$ and $f(b)$. That is, if $f(a)<f(b)$ and $y\in\left(f(a),f(b)\right)$ or if $f(a)>f(b)$ and $y\in\left(f(b),f(a)\right)$ then $\exists c\in\left(a,b\right)$ such that $f(c)=y$.

**Proof:**This is a nice proof, which you should know. It involves the facts that the connected sets in $\RR$ are the intervals and that the continuous image of a connected set is connected (both of which you should also be able to prove). **Definition:**Given a topological space $X$ and a point $x\in X$, the**connected component of $X$ containing $x$**, $C_x$, is the maximal connected subset of $X$ which contains $x$. Here,*maximal*means that if $A$ is a connected subset of $X$ such that $x\in A$, then $A\subset C_x$.**Theorem:**The term*connected component*is well-defined. That is, connected components exist and are unique. In fact, at the point $x$ in the topological space $X$, the connected component is given by $$C_x=\bigcup\left\{S\subset X \mid x\in S\ \text{and}\ S\ \text{is connected}\right\}\ \ .$$**Theorem:**The connected components in a topological space form a*partition*of that space. That is, if $C_1$ and $C_2$ are connected components, then either $C_1=C_2$ or $C_1\cap C_2=\emptyset$.**Definition:**A topological space is called**totally disconnected**if every point is a connected component.**Examples:**$\ZZ$, $\QQ$, and the Cantor set are totally disconnected.**Definition:**A point in a topological space is called a**cut point**if removing the point disconnects the space.**Examples:**Space # cut points # non-cut points $(a,b)$ $\infty$ 0 $[a,b]$ $\infty$ 2 $(a,b]$ or $[a,b)$ $\infty$ 1 $\left([-1,1]\times\{0\}\right)\cup\left(\{0\}\times[-1,1]\right)\subset\RR^2$ $\infty$ 4 $\RR$ $\infty$ 0 $\RR\sqcup\RR$ 0 $\infty$ $\RR^n$ for $n≥2$ 0 $\infty$ $S^1$ 0 $\infty$ **Definition:**In a topological space $(X,\Oo)$ a collection $\left\{O_\alpha \mid \alpha\in A\right\}$ is an**open cover**if $\left\{O_\alpha \mid \alpha\in A\right\}\subset\Oo$ and $X\subset\bigcup\left\{O_\alpha \mid \alpha\in A\right\}$. A subset $\left\{O_\alpha \mid \alpha\in B\right\}$ where $B\subset A$ is a**subcover**if it is still true that $X\subset\bigcup\left\{O_\alpha \mid \alpha\in B\right\}$.**Definition:**A topological space $X$ is called**compact**if every open cover of $X$ has a finite subcover; otherwise it is called**non-compact**.**Definition:**A topological space $X$ is called**locally compact**if every point of $X$ has a compact neighborhood.*[Note this is a "type I" use of the adverb "locally."]***Example:**For any $n\in\NN$, $\RR^n$ is non-compact.**Example:**$(0,1)\subset\RR$ is non-compact.*[After all, it is homeomorphic to $\RR$. ...also easy to prove directly.]***Theorem:**$\forall a,b\in\RR$ with $a<b$, $[a,b]\subset\RR$ is compact.

**Proof:**This is a nice proof that uses the following property of the real numbers:- The
**Least Upper Bound Property**Every nonempty, bounded set of real numbers has a least upper bound.

Recall that for any $n\in\NN$, a set $S\subset\RR^n$ is called**bounded**if $\exists r\in\RR$ such that $S\subset B(\vec{0},r)$. Also, given a set $S\subset\RR$, a**least upper bound**for $S$ is a number $L\in\RR$ such that- $L$ is an
**upper bound**for $S$, meaning: $\forall x\in S$, $x≤L$; and - if $L^\prime$ is any other upper bound for $S$, then $L≤L^\prime$.

- $L$ is an
**Example:**In an interesting contrast to what happened above with intervals of real numbers, $[0,1]\cap\QQ$ is non-compact.**Theorem:**A closed subset of a compact space is compact.**Theorem:**Let $f$ be a continuous function from a compact topological space $X$ to a topological space $Y$. If $f$ is onto then $Y$ is compact.**Corollary:**Let $X$ and $Y$ be homeomorphic topological spaces. Then $X$ is compact if and only if $Y$ is compact.**Corollary:**Let $f$ be a continuous function from a compact topological space $X$ to a topological space $Y$. Then $f(X)$ is a compact subspace of $Y$.*[This result is usually stated as "the continuous image of a compact set is compact."]***Proposition:**Let $C\subset\RR^n$ for some $n\in\NN$. If $C$ is compact then it is bounded.- Homeomorphism invariants we have to this point:
- connectedness
- compactness
- number of cut points
- number of non-cut points
- number of connected components

*F:**Content:***Theorem:**If $X$ and $Y$ are compact topological spaces then $X\times Y$ is compact.- The
**Heine-Borel Theorem:**A subset of $\RR^n$ is compact if and only if it is closed and bounded. **Proposition:**A compact subset of a Hausdorff space is closed.**Theorem:**A continuous, 1-1, and onto map $f:X\to Y$ from a compact topological space $X$ to a Hausdorff topological space $Y$ is a homeomorphism.**Definition:**Given any collection of topological spaces $\{(X_\alpha,\Oo_\alpha) \mid \alpha\in A\}$, we define the**Cartesian product**to be the topological space each of whose points is a sequence of points, each in one $X_\alpha$ (equivalently: a point is a function $x$ on $A$ such that $x(\alpha)\in X_\alpha$ $\forall\alpha\in A$ — think of $x(\alpha)$ as the*$\alpha$-coordinate*of the point $x$) and whose topology comes from the base consisting of sets like $\{x \mid x(\alpha)\in U_\alpha \forall\alpha\in A\}$ where $\{U_\alpha \mid \alpha\in A\}$ is a particular choice of sets satisfying $U_\alpha\in\Oo_\alpha$ $\forall\alpha\in A$ and $U_\alpha = X_\alpha$ for all but a finite number of $\alpha$'s. The notation for this Cartesian product is $\prod_{\alpha\in A} X_\alpha$.**Definition:**On the Cartesian product $\prod_{\alpha\in A} X_\alpha$ the function which sends a point $x$ (thought of as a function on $A$) to $x(\alpha)\in X_\alpha$, for some particular $\alpha\in A$, is called the**projection function**and written $\pi_\alpha$.**Theorem:**The topology we use on a Cartesian product is the coarsest one which makes all projections continuous.**Tychonoff's Theorem:**an arbitrary Cartesian product of compact spaces is compact.**Definition:**A**sequence**in a topological space $X$ is a function $x:\NN\to X$.*[One thinks of a sequence as like an infinitely long vector, $(x(1),x(2),\dots)$.]***Definition:**Given a sequence $x$ in a topological space $X$, a**subsequence**is another sequence $y$ formed as $y(i)=x(r(i))$ $\forall i\in\NN$, where $r:\NN\to\NN$ is a strictly increasing function (*i.e.,*$r(i+1)>r(i)$ $\forall i\in\NN$).**Definition:**A sequence $x$ in a topological space $(X,\Oo)$**converges to $L\in X$**if $\forall O\in\Oo$ $\exists N\in\NN$ such that $\forall k\in\NN$ satisfying $k≥N$, $x(k)\in O$. If such an $L$ exists, we say the sequence is**convergent**.**Definition:**A topological space is called**sequentially compact**if any sequence in the space has a convergent subsequence.**Theorem:**If $X$ is a compact topological space, then it is sequentially compact.**Theorem:**If $X$ is a sequentially compact topological space whose topology comes from a metric, then it is compact.**Definition:**If $X$ is a set, then a binary relation $\sim$ on $X$ (which means that $\forall x,y\in X$, $x\sim y$ is either true or false) is an**equivalence relation**if it satisfies:- $\forall x\in X$, $x\sim x$
*[this property is called "reflexivity"]* - $\forall x,y\in X$, $x\sim y$ if and only if
$y\sim x$
*["symmetry"]* - $\forall x,y,z\in X$, if $x\sim y$ and $y\sim z$, then
$x\sim z$
*["transitivity"]*

- $\forall x\in X$, $x\sim x$
**Definition:**Given a set $X$, an equivalence relation $\sim$ on $X$, and a point $x\in X$, the**equivalence class**of $x$ is the set $[x]=\{y\in X \mid y\sim x\}$.**Definition:**Given a set $X$, a**partition**of $X$ is a decomposition of $X$ into a collection of disjoint subsets whose union is all of $X$.**Proposition:**If $\sim$ is an equivalence relation on some set $X$, then the equivalence classes of $\sim$ form a partition of $X$.**Proposition:**If $X$ is a set with fixed partition, then the relation $\sim$ on $X$ defined by $x\sim y$ for $x,y\in X$ if and only if $x$ and $y$ are in the same partition subset, is an equivalence relation.**Definition:**Given a set $X$ with equivalence relation $\sim$, the set of equivalence classes is denoted $X/\sim$, which is pronounced "$X$ mod $\sim$." The map $q:X\to X/\sim:x\mapsto[x]$ here is called the**quotient map**.**Definition:**Given a topological space $(X,\Oo)$ whose set $X$ of points has some equivalence relation $\sim$, we define the**quotient topology**on $X/\sim$ to be the coarsest topology which makes the quotient map $q$ continuous.

Equivalently, the open sets of $X/\sim$ are exactly the sets of equivalence classes whose unions are open sets in $X$.**Examples:**Here are some of the many examples we can make with the quotient construction:- the
**torus** - the
**Klein bottle** - $S^1$, in several ways
- Let $X=\{\vec{x}\in\RR^3 \mid \vec{x}\neq\vec{0}\}$. On this
$X$, define the equivalence relation $\vec{x}\sim\vec{y}$ if
$\exists t\in\RR$ such that $\vec{x}=t\vec{y}$. [So, the
equivalence classes are lines through the origin (although
without the origin — we removed it first). In this case,
$X/\sim$ is called the
**real projective plane**and denoted $\RR\PP^2$ (or $\PP_\RR^2$ or even just $\PP^2$ if the real numbers are understood — but note that there is also a*complex projective plane!*).

- the

Do at least two of the following problems (due next class, which is Monday):*HW4*- Carefully read through the proof in Allen Hatcher's
Notes
on Introductory Point-Set Topology of the statement
*"If $X$ and $Y$ are compact topological spaces, then so is $X\times Y$,"*which is on p32. Now apply this proof to the particular case where $X=[0,17]\subset\RR$ and $Y=[0,\pi]\subset\RR$, so that $X\times Y$ is a closed $17\times\pi$ rectangle $R$ nestled into the corner of the first quadrant of the plane $\RR^2$.

On $R$ use the open cover consisting of all the sets $R\cap B$, where $B$ is any ball of radius $1/2^j$ for some $j\in\NN$ and centered at some point $(x,y)\in R$.

Now explain carefully how Hatcher's proof applies in this case. That is, for each variable he uses in his proof ($U_{xy}$, $V_{xy}$, $U_x$, $y_1, \dots, y_n$,*etc.*), tell what it is in this example and how it does what Hatcher says it does. - Can you think of a set in some topological space which is compact but not closed? You'll have to look at one of the more exotic spaces we've discussed in class. (E.g., by theorems we proved on Friday, this space will have to be non-Hausdorff.)
- Prove that a topological space $X$ is Hausdorff if and only if
the
*diagonal*$\Delta = \{(x,x)\in X\times X \mid x\in X\}$ is a closed subspace of $X\times X$.

- Carefully read through the proof in Allen Hatcher's
Notes
on Introductory Point-Set Topology of the statement

*M:**Content:***Example:**The unit ball in Hilbert space is not sequentially compact.**Proposition:**Suppose $X$ is a topological space on which there is an equivalence relation $\sim$, $Y$ is a topological space, and $f:X\to Y$ is a continuous function. If $f$ satisfies $f(x_1)=f(x_2)$ $\forall x_1,x_2\in X$ for which $x_1\sim x_2$, then the function $\overline{f}:X/\sim\to Y:[x]\mapsto f(x)$ is well-defined and continuous.

Conversely, given $X$, $\sim$, and $Y$, if $g:X/\sim\to Y$ is continuous, then the function $g\circ q:X\to Y$, where $q:X\to X/\sim$ is the quotient map, is continuous and obeys $\overline{g\circ q}=g$.

$$\begin{matrix} X&&\\ \downarrow&\searrow&\\ X/\sim&\rightarrow& Y\end{matrix}$$**Proposition:**Suppose $X$ is a topological space on which there is an equivalence relation $\sim$, $Y$ is a topological space, and $f:X/\sim\to Y$ is a function. Then $f$ is continuous if and only if $f\circ q:X\to Y$, where $q:X\to X/\sim$ is the quotient map.**Proposition:**Suppose $f:X\to Y$ is an onto continuous map from a compact topological space $X$ to a Hausdorff topological space $Y$. On $X$ define an equivalence relation by: $x,y\in X$ satisfy $x\sim y$ if $f(x)=f(y)$; in other words, the equivalence classes of $\sim$ are the sets $f^{-1}(\{y\})$ for all $y\in Y$. Then $\overline{f}:X/\sim\to Y$ is a homeomorphism.**Proposition:**Suppose $f:X\to Y$ is a function from a topological space $(X,\Oo_X$ to a set $Y$. Then the collection of sets $\Oo_Y=\{ O\subset Y \mid f^{-1}(O)\in\Oo_X\}$ is a topology on $Y$. This topology is the discrete topology on $Y\smallsetminus f(X)$.**Examples:**Two versions of $S^1$ as a quotient space, $[0,1]/\sim$ where $0\sim1$ is the only non-trivial equivalence, and $\RR/\ZZ$. The latter needs the following definition and proposition:**Definition:**Suppose $X$ is a set on which is defined some equivalence relation $\sim$. Then a set $S\subset X$ is said to be**saturated**if it is a union of equivalence classes. Equivalently, $S$ is saturated if $\forall s\in S$, if $x\sim s$ for some $x\in X$ then $x\in S$.**Definition:**A function $f:X\to Y$ between topological spaces is called an**open map**if it sends open sets in $X$ to open sets in $Y$.**Proposition:**Suppose $f:X\to Y$ is an onto continuous map from a topological space $X$ to a topological space $Y$. On $X$ define an equivalence relation by: $x,y\in X$ satisfy $x\sim y$ if $f(x)=f(y)$; in other words, the equivalence classes of $\sim$ are the sets $f^{-1}(\{y\})$ for all $y\in Y$. Then $\overline{f}:X/\sim\to Y$ is a homeomorphism if $f$ sends the saturated open sets to open sets. If, for example, $f$ is an open map, $\overline{f}$ will be a homeomorphism.**Examples:**Several constructions of $S^2$ as quotient spaces.**Definition:**Suppose $X$ and $Y$ are topological spaces, $C\subset X$ is closed, and $f:C\to Y$ is a homeomorphism of $C$ with a closed subset $f(C)\subset Y$. On $X\sqcup Y$ define the equivalence relation $\sim$ whose equivalence classes are all singletons with the exception of the pairs $\{c,f(c)\}$ for all $c\in C$; equivalently, let $c\sim f(c)$ for all $c\in C$ be the only non-trivial equivalences. Then**$X$ glued to $Y$ along $C$ by $f$**, written $X\cup_f Y$, is the quotient space $\left(X\sqcup Y\right)/\sim$.**Example:**Let $D_1$ and $D_2$ be two copies of the unit disk in $\RR^2$. Let $C\subset D_1$ be its boundary circle, and let $f:C\to D_2$ be the map which just sends the boundary circle to itself (the version of itself in the other disk). Then $D_1\cup_f D_2$ is homeomorphic to $S^2$.

*W:**Content:***Definition:**Given a topological space $X$, the**cone on $X$**, written $CX$, is the space $\left(X\times[0,1]\right)/\sim$ where $\sim$ is the equivalence relation on $X\times[0,1]$ whose equivalence classes are all of the single points $(x,t)$ where $x\in X$ and $t\in[0,1)$ and one class $\{(x,1) \mid x\in X\}$.**Example:**The cone on the circle, $CS^1$, is homeomorphic to the closed disk in $\RR^2$.**Definition:**Given a topological space $X$, the**suspension of $X$**, written $SX$, is the space $\left(X\times[0,1]\right)/\sim$ where $\sim$ is the equivalence relation on $X\times[0,1]$ whose equivalence classes are all of the single points $(x,t)$ where $x\in X$ and $t\in(0,1)$ and two other classes $\{(x,0) \mid x\in X\}$ and $\{(x,1) \mid x\in X\}$.**Examples:**The suspension of the $n$-sphere is homeomorphic to the $(n+1)$-sphere, $S\,S^n\cong S^{n+1}$.**Definition:**A**pointed topological space**is a topological space together with a choice of a particular point, often called the**basepoint**.**Definition:**Given two pointed topological spaces $(X,x_0)$ and $(Y,y_0)$, the**wedge**(or**wedge sum**) of these spaces, written $X\vee Y$, is the pointed topological space $\left(X\sqcup Y\right)/\sim$ where $\sim$ has equivalence classes of all isolated points in $X$ or $Y$ except for $x_0$ and $y_0$, and the additional class $\{x_0,y_0\}$*[i.e., the equivalence relation leaves every point of $X$ and $Y$ alone except it glues $x_0$ to $y_0$]*. The basepoint of $X\vee Y$ is the one non-trivial equivalence class;*i.e.,*it is the point where the basepoints $x_0$ and $y_0$ were glued together.**Definition:**Let $n_1,\dots,n_k$ be a finite collection of natural numbers. Then the topological space $S^{n_1}\vee \dots \vee S^{n_k}$ is called a**bouquet of spheres**(where each of the sphere is given any basepoint you like). If $n_1=\dots=n_k=1$, then the topological space $S^{n_1}\vee \dots \vee S^{n_k}$ is called a**bouquet of circles**.**Example:**If you puncture a torus and let the hole grow bigger and bigger, in the end you have a space which looks like a slightly thickened version of a bouquet of two circles.**Definition:**Given a topological space $X$, a**compactification of $X$**is a compact topological space $Y$ together with map $f:X\to Y$ which is a homeomorphism of $X$ with $f(X)$.**Definition:**Given a topological space $X$, the**one-point compactification of $X$**is the space $\overline{X}^{{}_1}$ whose points are all the points of $X$ and one additional point, usually denoted "$\infty$", with open sets:- all the open sets of $X$, and
- all sets of the form $K^c$, where $K$ is a compact subset of $X$ and the complement is taken in $\overline{X}^{{}_1}$; equivalently, take the complement of $K$ in $X$ and add also the extra point $\infty$.

**Theorem:**The one-point compactification of any topological space is a compactification of that space.*[I.e., the name is not inappropriate.]***Examples:**There are many other compactifications, often named after one or more famous mathematicians, useful both in particular circumstances where one wants the compactification to have some special nice structure continued from the interior out to the extra compactification points, and also in more general situations (but usually the general constructions are far more abstract and hard to work with or visualize; see the**Stone-Čech Compactification**, for example). For example:- The one-point compactification of $(0,1)\subset\RR$ is (homeomorphic to) the circle $S^1$.
- A two-point compactification of $(0,1)\subset\RR$ is $[0,1]\subset\RR$; in some sense, this preserves more of the structure of $(0,1)$, which seems to have two "ends".
- We know $(0,1)$ is homeomorphic to $\RR^1$, so in some sense the two compactifications of $\RR^1$ just mentioned are by gluing both positive and negative infinity together to make $S^1$, and another version which keeps those infinities separate, making $[0,1]$.
- The one-point compactification of the plane $\RR^2$ is the $2$-sphere $S^2$. This glues "all the infinities in all directions" into one point ... but that is actually the right thing to do in complex analysis, when thing of $\RR^2$ as the complex plane $\CC$.
- In any dimension, the closed unit ball
$\overline{B(\vec{0},1)}$ is a compactification of the open
unit ball $B(\vec{0},1)$.
*[This actually is what is going on in example 2, above.]*

**Definition:**Let $X$ be a topological space and suppose there exists an increasing sequence of subsets of $X$, so $K_1\subset K_2\subset\dots\subset X$, such that the interiors of the $K_j$'s cover $X$. Then make a choice for each $j$ of a connected component of $K_j^c$, call it $C_j$, in such a way that $C_1\supset C_2\supset\dots$. Such a sequence determines an**end**of $X$.**Definition:**A set $G$ with a binary operation $G\times G\to G:(g,h)\mapsto g\cdot h$ is called a**group**if- $\forall a,b,c\in G$ $(a\cdot b)\cdot c=a\cdot(b\cdot c)$
*[the associative law]* - $\exists e\in G$ such that $\forall g\in G$,
$e\cdot g=g\cdot e=g$
*[existence of an identity]* - $\forall g\in G$ $\exists g^{-1}\in G$ such that
$g\cdot g^{-1}=e$
*[existence of inverses]*.

- $\forall a,b,c\in G$ $(a\cdot b)\cdot c=a\cdot(b\cdot c)$
**Examples:**The following are all groups:- $\RR$ with the operation of addition
- $\RR\smallsetminus\{0\}$ with multiplication
- $\ZZ$ with addition
- Fix some $n\in\NN$, $\ZZ/n\ZZ$, pronounced
**$\ZZ$ mod $n$ $\ZZ$**, is the set of equivalence classes in $\ZZ$ under the equivalence relation $a\sim b$ if and only if $a-b$ is a multiple of $n$. The operation of multiplication $[a]\cdot[b]=[ab]$ makes $\ZZ/n\ZZ$ a group if and only if $n$ is prime. $\ZZ/n\ZZ$ is always a group under the operation of addition $[a]\cdot[b]=[a+b]$.

**Definition:**Given a group $G$, we say it is a**topological group**if $G$ is a topological space and both the multiplication map $G\times G\to G$ and the inversion map $G\to G:g\mapsto g^{-1}$ are continuous.**Definition:**Given a group $G$ and a set $X$, a**[left] group action of $G$ on $X$**is a map $G\times X\to X:(g,x)\mapsto g\cdot x$ satisfying:- $\forall a,b\in G$ and $x\in X$,
$(a\cdot b)\cdot x=a\cdot(b\cdot x)$
*[also called associativity]* - $\forall x\in X$, $e\cdot x=x$, where $e$ is the identity of $G$.

- $\forall a,b\in G$ and $x\in X$,
$(a\cdot b)\cdot x=a\cdot(b\cdot x)$
**Examples:**Here are some basic actions:- Any group acts on any space by the
**trivial action**, where $g\cdot x=x$ $\forall g,x$. This is not very interesting. - $\RR$ acts on $\RR^2$ by $t\cdot(x,y)=(x+t\sqrt{2},y-t\pi)$ (and many other actions).
- $\ZZ/12\ZZ$ acts on $\RR^2$ by rotation: Think of $\RR^2$ as $\CC$, the complex numbers. Then define the action by $[a]\cdot z=e^{ai\pi/6} z$. Note that this is well-defined, since if $b$ is any other integer such that $[a]=[b]$, that means that $a-b=12k$ for some $k\in\ZZ$, and thus $$e^{ai\pi/6}=e^{(b+12k)i\pi/6}=e^{(bi\pi/6)+(12ki\pi/6)}=e^{bi\pi/6}e^{12ki\pi/6}=e^{bi\pi/6}\left(e^{2\pi i}\right)^k=e^{bi\pi/6} 1^k=e^{bi\pi/6}$$ so in both $[a]\cdot x$ and $[b]\cdot x$ we are multiplying $x$ by the same complex number and so the action is the same.
- $\ZZ$ acts on $\RR$ by $n\cdot x=x+n$

- Any group acts on any space by the
**Definition:**Suppose the group $G$ acts on the space $X$. Then for $x\in X$, the**stabilizer of $x$**is the set $G_x=\{g\in G \mid g\cdot x = x\}\subset G$ and the**orbit of $x$**is $G\cdot x=\{g\cdot x \mid g\in G\}$.**Example:**In the above example of the action of $\ZZ/12\ZZ$ on $\RR^2$, all points other than the origin of $\RR^2$ have stabilizer which is just the identity element $[0]\in\ZZ/12\ZZ$, while the stabilizer of the origin is all of $\ZZ/12\ZZ$. The orbits of any hour marker on a clock dial centered at the origin of $\RR^2$ is the set of all hour markers; the orbit of, for example, any half hour marker is all half hour markers (but not the hour markers),*etc.*

*F:**Content:*- Apology about mistaken notation
*visavis*wedge and smash: in differential geometry, one takes the "wedge" of things called differential forms, written $\alpha\wedge\beta$; however, in topology, the*wedge sum*of two spaces $X$ and $Y$ is written $X\vee Y$, while the notation $X\wedge Y$ is used for the*smash product*of the spaces. **Definition:**Suppose an equivalence relation $\sim_X$ is defined on a set $X$. Given any $S\subset X$, we can define a relation $\sim_S$ by $s\sim_S t$, for $s,t\in S$, if and only if $s\sim_X t$. This $\sim_S$ is called**relation induced by $\sim$**, and sometimes we will denote the original and the induced relations by the same symbol $\sim$.**Proposition:**Suppose $X$ is a set with equivalence relation $\sim_X$, $S\subset X$, and $\sim_S$ is the induced relation on $S$. If every equivalence class in $X$ has at least one element in $S$ then $X/\sim_X$ is bijective with $S/\sim_S$. Furthermore, if $X$ is a topological space, then that bijection is a homeomorphism.**Definition:**Suppose $X$, $Y$, and $Z$ are topological spaces and there exist a 1-1 map $i:X\hookrightarrow Z$ and an onto map $p:Z\twoheadrightarrow Y$. Suppose further that $\forall y\in Y$, $\exists O\subset Y$ an open set containing $y$ and an homeomorphism $\varphi_O:p^{-1}(O)\to O\times X$ which satisfies $p(z)=\pi_1(\varphi_O(z))$ $\forall z\in p^{-1}(O)$, where $\pi_1:O\times X\to O:(o,x)\mapsto o$ is the projection onto the first factor.*[This property is called "local triviality".]*Then this scenario $X\hookrightarrow Z\twoheadrightarrow Y$ is called a**fibration**with**base space**$Y$,**total space**$Z$, and**fiber**$X$.**Definition:**Given topological spaces $X$ and $Y$, let $Z=X\times Y$. Fix some point $y_0\in Y$ and let $i:X\hookrightarrow Z:x\mapsto (x,y_0)$. Also let $p:Z\twoheadrightarrow Y$ be the projection $\pi_2:Z=X\times Y\to Y:(x,y)\mapsto y$. Then this $X\hookrightarrow Z\twoheadrightarrow Y$ is called the**trivial fibration over $Y$ with fiber $X$**.**Examples:**The cylinder $\RR\times S^1$ is a trivial fibration over $S^1$ with fiber $\RR$, while the Möbius band is a non-trivial fibration with the same fiber and base space.- Think of $S^3$ as the one-point compactification of $\RR^3$. Now
fill it with circles, as follows:
- One circle will be the $z$-axis, together with the point at infinity — this is then the one-point compactification of $\RR^1$, so it is a circle, homeomorphic to $S^1$.
- Another circle will be the unit circle in the $x$-$y$ plane.
- Next, fill up all the rest of space with layers of tori,
all of which have that unit circle in the $x$-$y$ plane as
their core. Make the inner edge of the tori pass through
the point $(x,0,0)$ for some $x\in\RR$ satisfying $0\lt x\lt1$
while the outer edge of that same torus passes through
$(1/x,0,0)$, so that the tori will fill all of space as $x$
ranges over the open unit interval.

Now fill each such torus with a family of circles which go once around diagonally

The collection of all circles on all tori is the rest of the circles we use to fill $S^3$.

*E.g.,*this one and this also are quite good.)

Define an equivalence relation $\sim$ on $S^3$ whose equivalence classes are the above circles. Notice that every single circle passes through the closed unit disk $D$ in the $x$-$y$ plane at exactly one point, except the edge, which is one entire circle. Therefore modding $D$ out by the induced equivalence relation $\sim$ on $D$ leaves everything alone except that it crushes the edge of $D$ to a single point, yielding $S^2$.

We can map $S^1$ 1-1 into $S^3$ as any of the above circles. Then $S^3$ has an onto map — the quotient map — to $S^3/\sim\cong D/\sim\cong S^2$. This diagram $$S^1 \hookrightarrow S^3\twoheadrightarrow S^2$$ is called the**Hopf fibration**. **Definition:**Let $G$ be a group and $H\subset G$. Suppose- $e\in H$, where $e$ is the identity element of $G$,
- $\forall h_1,h_2\in H$, $h_1\cdot h_2\in H$, and
- $\forall h\in H$, $h^{-1}\in H$,

**subgroup**of $G$.**Proposition:**Suppose $G$ is a group which acts on a space $X$ and $x\in X$. Then the stabilizer $G_x$ is a subgroup of $G$.**Examples:**Here are some more groups we will use:- Think of the 1-sphere $S^1$, sitting inside $\RR^2=\CC$, as the complex numbers of modulus 1. Then multiplication makes this a group: that is the group structure we will always use on $S^1$. Note that it can also be described as addition of angles.
- The set of invertible $n\times n$-matrices with real
coefficients is a group called the
**general linear group**$GL_n(\RR)$. - The set of $n\times n$-matrices with real coefficients and
determinant $1$ is a group called the
**special linear group**$SL_n(\RR)$. - The set of $n\times n$ orthogonal matrices,
*i.e.,*matrices whose columns form an orthonormal basis of $\RR^n$ is a group called the**orthogonal group**$O(n)$. Equivalently, $O(n)=\{A\in GL_n(\RR) \mid A\cdot A^t=Id\}$. - $SO(n)=SL_n(\RR)\cap O(n)$ is a group called the
**special orthogonal group**. - The set of $n\times n$-matrices with integral coefficients and determinant $1$ is a group written $SL_n(\ZZ)$.

**Example:**$SO(2)$ is homeomorphic to $S^1$.**Examples:**Some examples of actions with groups we know and of their corresponding quotient spaces....

- Apology about mistaken notation
Do at least two of the following problems (due next class, which is Monday):*HW5*- Suppose $X$ is a compact topological space. What is is its one-point compactification $\overline{X}^{{}_1}$? Give as simple a description as you can of $\overline{X}^{{}_1}$ and prove that your description is correct.
- We talked about two compactifications of the open interval
$(0,1)\subset\RR$: the one-point compactification, which we
said was homeomorphic to $S^1$, and a "two-point"
compactification as $[0,1]$.

Since $(0,1)$ is homeomorphic to all of $\RR$, there should be two corresponding compactifications of $\RR$: the one-point compactification $\overline{\RR}^{{}_1}$ (which we did discuss in class and talk about how it is homeomorphic to $S^1$ — think stereographic projection!), and a two-point compactification which will will denote (in this problem only, it is not a standard notation) $\overline{\RR}^{{}_2}$.- Describe $\overline{\RR}^{{}_2}$ carefully: what are its points, and what are all it's open sets. Note: using Heine-Borel, you can describe the open sets in $\overline{\RR}^{{}_2}$ quite simply.
- Suppose $f:\RR\to\RR$ is a continuous function. When can
you extend $f$ to be a continuous function whose domain and
codomain is $\overline{\RR}^{{}_1}$ or $\overline{\RR}^{{}_2}$?
That is, what conditions on $f$ tells you when there will exist
continuous functions
$$\begin{align*}
\overline{f}^{{}_{11}}:&\overline{\RR}^{{}_1}\to\overline{\RR}^{{}_1}\\
\overline{f}^{{}_{12}}:&\overline{\RR}^{{}_1}\to\overline{\RR}^{{}_2}\\
\overline{f}^{{}_{21}}:&\overline{\RR}^{{}_2}\to\overline{\RR}^{{}_1}\\
\overline{f}^{{}_{22}}:&\overline{\RR}^{{}_2}\to\overline{\RR}^{{}_2}
\end{align*}$$
each of which equals $f$ on the points of $\RR$ but is extended
— explain how! — to also have values at the extra
point(s) of the compactifications; those values are allowed to
be regular real numbers, or could even be (one of) the extra
value(s) in the compactification that is the codomain.

Prove your answer, and give an example of each type of function, and an example where the extension**does not**exist. - Suppose $p(x)$ is a real polynomial of a real variable.
Apply the conditions you just found for general function to
describe which such $p$ have which kinds of extensions.
*[Your answer will probably be in terms of properties of $p$ like its degree, leading coefficient, ....]* - How about a rational function? Rational functions have two
issues: what they do as $x\to\pm\infty$; and what they do near
vertical asymptotes. Can you figure out how to extend a
rational function $r(x)$ to a continuous function
$\overline{r}^{{}_{01}}:\RR\to\overline{\RR}^{{}_1}$ or a
continuous
$\overline{r}^{{}_{02}}:\RR\to\overline{\RR}^{{}_2}$ —
that is, don't worry about $x\to\pm\infty$, but extend $r$
across each of the vertical asymptotes (where a traditional
rational function is of course undefined due to division by
0)? When would each be possible or not?
*[Hint: think about some basic examples, like $1/x$ or $1/x^2$.]*

- Consider the group $\QQ$ (with addition being the group operation)
acting on the real line $\RR$. What is the quotient space
$\RR/\QQ$, in the sense: what are its points, and what is the
topology on those points?
*[Hint: the points are not particularly nice, but the topology is very simple.]*

*M:**Content:***Proposition:**Sometimes one writes $H\le G$ to indicate that $H$ is a subgroup of $G$. Using that notation:- $SO(n)\le SL_n(\RR)\le GL_n(\RR)$
- $SO(n)\le O(n)\le GL_n(\RR)$
- $SL_n(\ZZ)\le SL_n(\RR)$

**Definition:**Given a group $G$ and subgroup $H$, a**left coset of $H$**is a subset of the form $gH=\{gh \mid h\in H\}$ for some $g\in G$. Likewise, a**right coset of $H$**is a subset of the form $Hg=\{hg \mid h\in H\}$ for some $g\in G$. The set of left cosets of $H$ is denoted $G/H$, which is pronounced "$G$ mod $H$."**Definition:**A group $G$ is called**Abelian**if $\forall g,h\in G$, $gh=hg$.**Definition:**Given a group $G$ and subgroup $H$, we say $H$ is**normal**if $\forall g\in G$ and $\forall h\in H$, $ghg^{-1}\in H$.**Proposition:**If $G$ is a normal subgroup then the operation $g_1 H\cdot g_2 H = (g_1 g_2) H$ makes $G/H$ into a group, called the**quotient group**.**Definition:**Let $G$ and $H$ be groups. Then a map $\varphi:G\to H$ is a**homomorphism**if $\forall g_1,g_2\in G$, $\varphi(g_1\cdot g_2)=\varphi(g_1)\cdot\varphi(g_2)$.*[Note that in that equation, the first "$\cdot$" was multiplication in $G$ and the second was multiplication in $G$!]***Definition:**Starting with the symbols $a, b, c, \dots$, a**word**is a finite sequence of those symbols, and the corresponding symbols $a^{-1}, b^{-1}, c^{-1}, \dots$, in some order; note we consider the empty sequence also a word. A**reduced word**is one in which no symbol $x$ ever occurs immediately next to the symbol $x^{-1}$. The**free group on $a, b, c, \dots$**, written $F(a, b, c, \dots)$, is the group whose elements are all reduced words, whose group operation is concatenation of words (followed by making the resulting word reduced by removing adjacent pairs of symbols $x$ and $x^{-1}$), and whose identity is the empty word.**Example:**With only one symbol, the free group $F(a)$ is really just the group $(\ZZ,+)$: every element of $F(a)$ is some number of $a$'s next to each other, or some number of $a^{-1}$'s, or is empty, and that number — positive for the $a$'s, negative for the $a^{-1}$'s and zero for the empty word — is the corresponding element of $\ZZ$.**Example:**Already with two symbols, the free group $F(a,b)$ is a complicated, non-Abelian group: $ab$ is a different word from $ba$, so the words $a$ and $b$ do not commute!**Definition:**Give a set of words $w_1, \dots, w_n\in F(a, b, c, \dots)$, let $N$ be the smallest normal subgroup of $F(a, b, c, \dots)$ which contains these words $w_1, \dots, w_n$. Then the quotient group $F(a, b, c, \dots)/N$ is called the**group with generators $a, b, c, \dots$ and relations $w_1=id, \dots, w_n=id$**, sometimes written $F(a, b, c, \dots)/\left<w_1, \dots, w_n\right>$.**Example:**In the group $G=F(a,b)/\left<aba^{-1}b^{-1}\right>$, the classes of the elements $a$ and $b$ do commute, so really $G$ consists of a copy of $F(a)$ together with a copy of $F(b)$, which means it is really just two copies of $\ZZ$,*i.e.,*$G=\ZZ^2$.**Definition:**Given two topological spaces $X$ and $Y$ and continuous maps $f:X\to Y$ and $g:X\to Y$, we say**$f$ is homotopic to $g$**, written $f\simeq g$, if there exists a continuous map $F:X\times I \to Y$ (where $I$ is always the unit interval $[0,1]$ in homotopy theory) such that $\forall x\in X$, $F(x,0)=f(x)$ and $F(x,1)=g(x)$.**Definition:**Given two paths $\alpha$ and $\beta$ in a topological space $X$ which satisfy $\alpha(0)=\beta(0)=x_0$ and $\alpha(1)=\beta(1)=x_1$, a**path homotopy**between them is a continuous function $F:I\times I\to X$ such that $\forall s\in I$, $F(s,0)=\alpha(s)$ and $F(s,1)=\beta(s)$ and $\forall t\in I$, $F(0,t)=x_0$ and $F(1,t)=x_1$. If such a path homotopy exists, we write $\alpha\simeq_p\beta$.**Definition:**Given a pointed topological space $(X,x_0)$, a**loop at $x_0$**is a path in $X$ which starts and ends at $x_0$.**Definition:**Given two loops $\alpha$ and $\beta$ at $x_0$ in a pointed topological space $(X,x_0)$, we define the**composition of loops**as the loop $$(\alpha*\beta)(t)=\begin{cases} \alpha(2t) & \text{if }0≤t≤1/2\\ \beta(2t-1) & \text{if }1/2≤t≤1\end{cases}\ .$$**Proposition:**Given loops $\alpha_0$, $\alpha_1$, $\beta_0$, and $\beta_0$ at $x_0$ in a pointed topological space $(X,x_0)$, if $\alpha_0\simeq_p\alpha_1$ and $\beta_0\simeq_p\beta_1$, the $\alpha_0*\beta_0\simeq_p\alpha_1*\beta_1$.**Definition:**Let $(X,x_0)$ be a pointed topological space. We define**the fundamental group $\pi_1(X,x_0)$ of $(X,x_0)$**to be the group whose elements are path homotopy equivalence classes of loops at $x_0$, whose group operation is $[\alpha]\cdot[\beta]=[\alpha*\beta]$, and whose identity element is the**trivial**[constant] path $e:I\to X:t\mapsto x_0$.**Theorem:**The fundamental group is a group.

**Proof:**Use a diagram like this one to prove associativity, and similar ones to handle inverses,*etc.***Proposition:**The fundamental group of $\RR^n$ is trivial $\forall n\in\NN$.

*W:**Content:***Definition:**Given groups $G$ and $H$ and a homomorphism $f:G\to H$, we say $f$ is an**isomorphism**if there exists another homomorphism $g:H\to G$ such that $f\circ g=id_H$ and $g\circ f=id_G$. If an isomorphism exists between two groups $G$ and $H$, we say they are**isomorphic**and write $G\cong H$.**Definition:**Given two topological spaces $X$ and $Y$, a continuous map $f:X\to Y$ is called a**homotopy equivalence**if there exists another continuous map $g:Y\to X$ such that $f\circ g\simeq id_H$ and $g\circ f\simeq id_G$ (recall $\simeq$ means "homotopic").**Definition:**If there exists a homotopy equivalence between two topological spaces, we say they have the same**homotopy type**.**Definition:**If a topological space has the homotopy type of a point, we say it is**contractible**.**Proposition:**A contractible topological space is path-connected.**Definition:**A pointed topological space $(X,x_0)$ is said to be**simply connected**if $X$ is path connected and $\pi_1(X,x_0)$ is the trivial group.**Proposition:**If a topological space is contractible then it is simply connected.**Example:**For any $n\in\NN$, $\RR^n$ is simply connected.**Theorem:**If $X$ and $Y$ are homotopy equivalent, path-connected spaces, then $\pi_1(X,x_0)\cong\pi_1(Y,y_0)$ for any $x_0\in X$ and $y_0\in Y$.**Theorem:**If $X$ is a path-connected topological space, then for any two points $x_0,x_1\in X$, $\pi_1(X,x_0)\cong\pi_1(X,x_1)$.**Definition:**A pair of topological spaces together with an onto map $p:E\twoheadrightarrow B$ is called a**covering space**if $\forall b\in B$ there exists and open $U\subset B$ such that $p^{-1}(U)$ is homeomorphic to $\sqcup_{\alpha\in A} V_\alpha$, where $A$ is some set and $p$ is a homeomorphism from each $V_\alpha$ to $U$. The space $B$ in the covering is called the**base space**while the space $E$ is called the**total space**.**Example:**We have seen that $\RR^1/\ZZ$ is homeomorphic to $S^1$, and the quotient map $p:\RR^1\to\RR^1/\ZZ$ is a covering of the base space $S^1$ with total space $\RR$.**Theorem:**The curve-lifting property of covering spaces.**Theorem:**The homotopy-lifting property of covering spaces.**Theorem:**Let $x_0$ be any point on $S^1$. Then $\pi_1(S^1,x_0)\cong\ZZ$.**Definition:**Suppose $(X,x_0)$ and $(Y,y_0)$ are pointed spaces. Then a**pointed map**$f:X\to Y$ is a map such that $f(x_0)=y_0$.**Definition:**Suppose $(X,x_0)$ and $(Y,y_0)$ are pointed topological spaces and $f:X\to Y$ is a pointed continuous map between them. Then the**induced map on fundamental groups**is the map $f_*:\pi_1(X,x_0)\to\pi_1(Y,y_0)$ defined by $f_*([\alpha])=[f\circ\alpha]$, where $\alpha$ is a loop in $X$ based at $x_0$.

*F:**Content:***Proposition:**If path-connected $X$ is simply connected, that for every $x_0,x_1\in X$, there exists a unique path-homotopy class of paths which go from $x_0$ to $x_1$.*[I.e., it is path connected in a very simple way....]***Brouwer's Fixed Point Theorem:**If $\overline{B(0,1)}$ is the closed unit ball in $\RR^2$, then any continuous map $f:\overline{B(0,1)}\to\overline{B(0,1)}$ has a**fixed point**, that is, $\exists x\in\overline{B(0,1)}$ such that $f(x)=x$.**The Fundamental Theorem of Algebra:**Any non-constant complex polynomial has zeros.**Definition:**Given groups $G$ and $H$, the**Cartesian product $G\times H$**is the group with elements $\left\{(g,h) \mid g\in G, h\in H\right\}$, group operation $(g_1,h_1)\cdot(g_2,h_2)=(g_1\cdot g_2,h_1\cdot h_2)$*[where the first "$\cdot$" there was in $G\times H$, the second in $G$, and the third in $H$]*, and identity $(id_G,id_H)$.**Theorem:**Given pointed topological spaces $(X,x_0)$ and $(Y,y_0)$, $\pi_1(X\times Y, (x_0,y_0))$ is isomorphic to $\pi_1(X,x_0)\times\pi_1(Y,y_0)$.**Definition:**We use the notation $\TT^1$ for the torus, $S^1\times S^1$.**Example:**$\pi_1(\TT^1)\cong\pi_1(S^1)\times\pi_1(S^1)=\ZZ\times\ZZ$.**Theorem:**For any $n\in\NN$, $n≥2$, $S^n$ is simply connected.**Definition:**Given pointed topological spaces $(X,x_0)$ and $(Y,y_0)$, a**pointed map from $X$ to $Y$**is a map $f:X\to Y$ which satisfies $f(x_0)=y_0$.**Definition:**Given pointed topological spaces $(X,x_0)$ and $(Y,y_0)$, and two pointed maps $f$ and $g$ from $X$ to $Y$, a**pointed homotopy**between $f$ and $g$ is a map $F:X\times I\to Y$ such that $\forall x\in X$, $F(x,0)=g(x)$ and $F(x,1)=g(x)$, and $\forall t\in I$, $F(x_0,t)=y_0$.**Definition:**Given pointed topological spaces $(X,x_0)$ and $(Y,y_0)$, we use the confusing notation $[X,Y]$ to represent the set of all pointed homotopy classes of pointed continuous maps from $(X,x_0)$ to $(Y,y_0)$.**Definition:**Given a pointed and path-connected topological space $(X,x_0)$, its**higher homotopy groups**are the groups $\pi_k(X,x_0)=\left[S^k,X\right]$, where the group operation is defined for $[\alpha],[\beta]\in\left[S^k,X\right]$ by $[\alpha]\cdot[\beta]=[\alpha*\beta]$, where we think of $S^k$ as $I^k/\sim$ where the relation $\sim$ collapses the whole boundary of $I^k$ to a point (being the basepoint), and so $\alpha$ and $\beta$ are maps from $I^k$ to $X$ which send the whole boundary to $x_0$, and $$(\alpha*\beta)(t_1,\dots,t_k)=\begin{cases} \alpha(2t_1,t_2,\dots,t_k) & \text{if }0≤t_1≤1/2\\ \beta((2t_1-1),t_2,\dots,t_k) & \text{if }1/2≤t_1≤1\end{cases}\ .$$**Theorem:**The higher homotopy groups are Abelian.

**Proof:**Look at this picture.**Example:**Consider the pointed topological space $(S^1\vee S^1,\overline{1})$ (which looks like a figure-eight). Call going around one loop in the clockwise direction $R$ and going around the other $L$. Then a simply connected covering space $p:E\twoheadrightarrow S^1\vee S^1$ can be constructed as follows: over the basepoint $\overline{1}$ put a vertex $e$. Corresponding to traversing each of $R$, $R^{-1}$ (which is just $R$ backwards), $L$, and $L^{-1}$, and labelled with that symbol, attach a line segment to $e$ which will be carried by $p$ homeomorphically to one of the $S^1$'s in one of the two directions, as specified by the label. Now continue indefinitely, adding vertices and, at each vertex, three new segments labelled by the symbols which were not used on the label of the previous segment. (Here is an image of this space, although this image uses a different labeling convention: each*vertex*is labelled by the sequence of labels we just described on the sides along the unique path from $e$ to that vertex; also, use $a$ instead of $R$ and $b$ instead of $L$.)

**Conclusion:**$\pi_1((S^1\vee S^1,\overline{1})$ is isomorphic to the free group $F(R,L)$.**Definition:**Given two groups $G$ and $H$, we define the**free product group**$G*H$ as the group consisting of**reduced words from $G$ and $H$**, where a word is a sequence of elements from $G$ and $H$, and reduced means removing every occurrence of $id_G$ or $id_H$ and replacing any occurrence of two elements from $G$ or $H$ next to each other by the single element from $G$ or $H$ which is their product in that group. The group operation is by concatenating words and then reducing.**Example:**$F(a,b)\cong F(a)*F(b)\cong\ZZ*\ZZ$*[which, notice,*. It follows that $\pi_1((S^1\vee S^1,\overline{1})\cong\pi_1(S^1,1)*\pi_1(S^1,1)$.**is not**$\ZZ^2$]

- There are a number of good sources to read about what we have been doing this week, including Wikipedia and Chapter 1 of Allen Hatcher's book Algebraic Topology.
Do at least three of the following problems (due next class, which is Monday):*HW6*- (a) Construct a continuous map which is 1-1 and onto
from the half-open, half-closed interval in $\RR$ to $S^1$; that
is, $f:[0,b)\to S^1$.

(b) Give three explanations that your map is not a homeomorphism: first, show directly that its inverse is not continuous. Second, use compactness, cut points, or something else from last week or before. Third, use fundamental groups. - How about a
**lower**homotopy group?

Our definition of the $n$-sphere even works for $n=0$: describe $S^0$.

Now, pick any point of $S^0$ as a basepoint $s_0$ to make it a pointed topological space, and let $(X,x_0)$ be any other pointed topological space. Describe $\pi_0(X,x_0)=\left[S^0,X\right]$. - We gave a definition above of the real projective plane
$\RR\PP^2$, but here is a new one: on $S^2$ define the equivalence
relation $\sim$ which has equivalence classes that are pairs of
points $\{p,-p\}$, where $-p$ is the point directly on the opposite
side of the sphere from the point $p$ ($-p$ is called the
**antipodal**point of $p$). The $\RR\PP^2$ can also be thought of as $S^2/\sim$; use the symbol $p$ to denote the quotient map $p:S^2\twoheadrightarrow\RR\PP^2$.

(a) Explain why this gives the same space as the definition we've used before for $\RR\PP^2$.

(b) Find a covering space for $\RR\PP^2$. Use it to compute $\pi_1(\RR\PP^2,x_0)$ where $x_0$ is the point of $\RR\PP^2$ consisting of the north and south poles of $S^2$.

(c) Let $\alpha$ be the semicircle on $S^2$ going from the north pole to the south pole. What is the class $[p\circ\alpha]\in\pi_1(\RR\PP^2,x_0)$? What happens when we square that element in $\pi_1(\RR\PP^2,x_0)$? - Some computations — give as complete an explanation of
your reasoning (a proof would be nice!) as you can:
- What is the fundamental group of the Möbius strip?
- Give an example of a simply connected space which is not contractible.
- Give an example of a topological space which is not homeomorphic to $S^1\vee S^1$ but which has fundamental group which is isomorphic to $F(R,L)$.
- What are the fundamental groups of the upper-case letters of the alphabet, viewed as subsets of $\RR^2$? Make a table! ...Hmm, which font you use might matter, so use this one: $$ABCDEFGHIJKLMNOPQRSTUVWXYZ$$ It's convenient that these letters are all connected ...

- Some more computations — again, please give as complete
an explanation as you can:
- What is the fundamental group of the wedge of $n$ circles?
*[Hint: try a direct generalization of what we did in class!]* - What is the fundamental group of $S^1\vee S^2$?
- Now generalize those two pieces: what is the fundamental group of a bouquet of spheres (definition above, here).

- What is the fundamental group of the wedge of $n$ circles?

- (a) Construct a continuous map which is 1-1 and onto
from the half-open, half-closed interval in $\RR$ to $S^1$; that
is, $f:[0,b)\to S^1$.

*M:**Content:*- Discussion of some of
*HW6*. **Examples:**We've seen how $\pi_1(\RR\RR^2)=\ZZ/2\ZZ$, how about another space with a finite fundamental group (the same one)? Think of $SO(3)$ as the rotations of three-dimensional space, do the arm-twisty thing to show a non-trivial loop of rotations, square it to show that that non-trivial element, when squared, is the identity. Hence $\pi_1(SO(3))\cong\ZZ/2\ZZ$. Here and here are demonstrations of the "cup trick." You can also see a version of the "Dirac Spinor Spanner" here.**Definition:**Given three groups $G$, $H$, and $A$ and two homorphisms $\varphi_G:A\to G$ and $\varphi_H:A\to H$, we define the**amalgamated free product $G*_A H$**as the quotient of the free product $G*H$ by the smallest normal subgroup which contains all elements of the form $\varphi_G(a^{-1})\varphi_H(a)$.*[The idea is that we are taking the free product and making the image of $A$ by $\varphi_G$ behave the same as its image by $\varphi_H$.]***The Seifert-Van Kampen Theorem:**Let $U$ and $V$ be open sets in a topological space $X$. Suppose $U\cup V=X$ and $U$, $V$, and $U\cap V$ are all path connected. Use as basepoint for $X$ a point $x_0\in U\cap V$. Looking at the inclusion maps $i^U:U\cap V\hookrightarrow U$ and $i^V:U\cap V\hookrightarrow V$ and let the induced maps on fundamental groups $i^U_*:\pi_1(U\cap V,x_0)\to\pi_1(U,x_0)$ and $i^V_*:\pi_1(U\cap V,x_0)\to\pi_1(V,x_0)$ be the homorphisms which define the amalgamated free product. Then $$\pi_1(X,x_0)\cong\pi_1(U,x_0)*_{\pi_1(U\cap V,x_0)}\pi_1(V,x_0)\ .$$**Theorem:**Using the notation of the definition above of the amalgamated free product: Suppose $G={e}$ is the trivial group. Then the free product is quite simple: $G*H\cong H$. If, also, $H=F(a, b, c, \dots)$ and we set $w_1=\varphi_H(a_1),\dots,w_n=\varphi_H(a_n)$, where $A=\left\{a_1,\dots,a_n\right\}$, then $\{e\}*_A H\cong F(a, b, c, \dots)/\left<w_1, \dots, w_n\right>$, the group with generators $a, b, c, \dots$ and relations $w_1=id, \dots, w_n=id$**Examples:**Three applications of the Van Kampen Theorem:- Let $X=S^2$, $U$ be a slightly extended northern hemisphere, and $V$ be a slightly extended southern hemisphere, so that $U\cap V\simeq S^1\times(0,1)$. Then $\pi_1(S^2)=\pi_1(U\cup V)\simeq\{e\}*_\ZZ\{e\}\cong\{e\}$, (an amalgamated product of trivial groups is trivial!) so $S^2$ is simply connected (again).
- Let $X=\TT^1$ be the torus, $U$ the punctured torus, and $V$ a disk to cover the puncture. Then $U\simeq S^1\vee S^1$ and $U\cap V\simeq S^1$, so $\pi_1(U)=F(a,b)$, $\pi_1(V)=\{e\}$, and $\pi_1(U\cap V)\cong\ZZ$. By the theorem immediately above and Van Kampen's Theorem (and calculations with the homomorphisms which define the amalgamated fee product), we have that $\pi_1(\TT^1)$ is the group with generators $\{a,b\}$ and relation $aba^{-1}b^{-1}$, which we have seen previously is isomorphic to $\ZZ^2$.
- Let $X=\Sigma_2$ be the two-holed torus, $U$ the punctured, two-holed torus, and $V$ a disk to cover the puncture. Then $U\simeq S^1\vee S^1\vee S^1\vee S^1$ (see, for example, this image) and $U\cap V\simeq S^1$, so $\pi_1(U)=F(a,b,c,d)$, $\pi_1(V)=\{e\}$, and $\pi_1(U\cap V)\cong\ZZ$. This time the homomorphism in the amalgamated free product takes $1\in\ZZ$ to $r=aba^{-1}b^{-1}cdc^{-1}d^{-1}$, so $\pi_1(\Sigma_2)$ is the group with generators $\{a,b,c,d\}$ and relation $r$.

**Definition:**Given a group $G$ acting on a space $X$, we say that the action is**transitive**if the orbit of some point $x_0\in X$ is all of $X$.*[Then the orbit of***any**point will be all of $X$.]**Proposition**If $G$ acts transitively on a space $X$ and the stabilizers are trivial, then choosing any $x_0\in X$ gives a 1-1, onto identification of $X$ with $G$ by sending $x\in X$ to the (unique) element $g\in G$ such that $x=g x_0$.**Definition:**A covering space $p:E\twoheadrightarrow B$ is a**universal covering space**of $B$ if the total space $E$ is simply connected.**Theorem:**The fiber $p^{-1}(b_0)$ of the universal covering space $p:E\twoheadrightarrow B$ over the basepoint $b_0\in B$ can be identified with the fundamental group of the base $B$.*[This gives the main way we have, other than the Van Kampen Theorem, of computing a fundamental group: find the universal covering space and look at the fiber over the base space.]***Example:**$p:\RR^2\twoheadrightarrow\RR^2/\ZZ^2=\TT^2$ is a covering of the torus, so $\RR^2$ is the universal covering space of $\TT^2$.- We shall try to make a similar example to this process of getting the universal cover of the square with identifications as $\RR^2$ also in the case of the octagon with identifications, which should get us the universal covering space of the two-holed torus $\Sigma_2$. Pictures like this one give a hint that the universal cover might be the disk. The problem will be to find the analogue of the $\ZZ^2$ and how it acts on the disk.

- Discussion of some of

*W:**Content:***Definition:**The points of theare the set $\Hh^2=\{(x,y)\in\RR^2 \mid y\gt0\}$.*[upper half-plane model of the]*hyperbolic plane**Definition:**A curve $\alpha:[0,1]\to\Hh^2$ of the form $\alpha(t)=(x(t),y(t))$ has**hyperbolic length**$$\Ll(\alpha)=\int_0^1{\frac{\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}}{y(t)}\,dt}$$**Definition:**Given two points $a,b\in\Hh^2$, the**hyperbolic distance**$d_\Hh(a,b)$ between the points is the shortest hyperbolic length of curves from $a$ to $b$. This distance function is also called the**hyperbolic metric**.**Theorem:**The hyperbolic metric makes $\Hh^2$ a metric space, and $\Hh^2$ with this metric topology is homeomorphic to $\RR^2$ (with, as usual, the Euclidean metric).**Definition:**A**hyperbolic geodesic**in $\Hh^2$ is a curve $\alpha:[0,1]\to\Hh^2$ such that $\forall s,t\in[0,1]$ with $s\lt t$, $d_\Hh(\alpha(s),\alpha(t))=\Ll(\alpha|_{[s,t]})$, where $\alpha|_{[s,t]}$ is the restriction of $\alpha$ to $[s,t]$.*[That is, a geodesic is the curve which minimizes length between all of its points.]***Proposiiton:**The geodesics in $\Hh^2$ are exactly the vertical lines and the semicircles with centers on the real axis. All of these geodesics are of infinite length, so that, in particular, from within the hyperbolic plane, the "edge" of hyperbolic space which we see with our Euclidean eyes, being the real axis in $\RR^2$, is infinitely far away when you look with hyperbolic eyes.**Definition:**Two hyperbolic geodescis are said to be**parallel**if they never intersect.**Theorem:**If "straight line" is taken to mean "hyperbolic geodesic", "point" is "point", and an angle between straight lines is defined as the Euclidean angle between the tangents to those hyperbolic geodesics at the point of intersection, then this system satisfies all of the axioms of Euclid's geometry except the parallel postulate. In fact, given any line and a point not on that line, there exist an infinite number of lines through the given point and parallel to the given line ... in hyperbolic geometry.**Definition:**Given a metric space $(X,d)$, a function $f:X\to X$ is called an**isometry**if $\forall x_1,x_2\in X$, $d(x_1,x_2)=d(f(x_1),f(x_2))$.- isometry group of $\Hh^2$, types of elements (parabolic, elliptic, and hyperbolic) and their actions
- hyperbolic horns (cylinders with the hyperbolic metrics)
**Definition:**A sphere is said to be a**Riemann surface of genus zero**, which a torus with $g\ge 1$ holes is said to be a**Riemann surface of genus $g$**.- polygons in $\Hh^2$ with isometries to identify sides, yielding Riemann surfaces of genus $g\ge 2$, or a punctured torus.
- Many beautiful expositions of this material exist on the 'net and in books. See, for example, Hyperbolic Geometry: The First 150 Years, by John Milnor, or Fuchsian Groups by Svetlana Katok.

*M:**Content:***Definition:**Suppose a group $G$ acts on a space $X$. A subset $A\subset X$ is called**$G$-invariant**if $\forall g\in G$, $\forall a\in A$, $ga\in A$.*[So every element of $G$ moves the set $A$ to itself — the individual points of $A$ may be moved by $G$, but all of $G$ stabilizes $A$***as a set**.]**Proposition:**Suppose a group $G$ acts on a space $X$ and $p:X\twoheadrightarrow X/G$ is the quotient map. Then for any $A\subset X/G$, the set $p^{-1}(A)$ is $G$-invariant.**Example:**Let $X$, $Y$, and $Z$ be the $x$-, $y$-, and $z$-axes, respectively, in $\RR^3$. Then $\pi_1\left(\RR^3\smallsetminus(X\cup Y\cup Z)\right)=F(a,b,c,d,e)$.*[Use homotopy equivalences between $\RR^3\smallsetminus(X\cup Y\cup Z)$ and $S^2$ minus six points, then with that and $\RR^2$ minus five points, then with that and the wedge of five circles.]***Example:**Let $\sim$ be the equivalence relation on $S^2$ all of whose equivalence classes are singletons except the class $\{N,P\}$, where $N$ is the north pole and $S$ the south. Then $\pi_1(S^2/\sim)\cong\ZZ$.*[Use the Van Kampen Theorem with halves of $S^2/\sim$ made by cutting it in half vertically.]***Definition:**A topological space $X$ is called**semi-locally simply connected**if $\forall x\in X$ there is an open neighborhood $O$ of $x$ such that every loop in $O$ is homotopic in $X$ to the trivial loop. That is, if $i:O\hookrightarrow X$ is the inclusion map, then $i^*:\pi_1(O)\to\pi_1(X)$ is the map which sends everything to the identity.**Examples:**- The Hawaiian earring is not semi-locally simply connected: any open neighborhood of the point where all the circles are tangent contains infinitely many circles and loops around those circles are not homotopically trivial, even in the whole space.
- The cone on the Hawaiian earring is simply connected, therefore it is semi-locally simply connected. It is, however, not locally simply connected since a small neighborhood of the point where all the circles are tangent, but not near the cone point, contains no simply connected smaller open neighborhood.

**Proposition:**A topological space which is has a universal covering space must be semi-locally simply connected.**Theorem:**A topological space $X$ which is connected, locally path-connected, and semi-locally simply connected has a universal covering space. That space can be described as follows: fix a basepoint $x_0\in X$. Let $\tilde{X}$ be the set of path-homotopy classes of paths starting at $x_0$. The covering map $p:\tilde{X}\to X$ is given by sending a class of curves to the endpoint of any representative.

*W:**Content:*- OK, it's the Hawaiian
**earring**, not "necklace". - correction of the definition of "locally simply connected", examples
- Students should hand in the written portion of their final
projects. This should be a 4/5 page document (or more, if you like;
much less would be odd) describing your resent research project.
It should include the following parts:
- A short introduction (maybe a paragraph or two?), giving an idea of the topic, where you will go with it and naming the high point you will get to by the end.
- Clear, formal definitions (see above in this web-page for
examples; also see this handout from an old class
which discusses important features in a good definition) of
**all**of the (new — no need to repeat definitions from this class) technical terms you will mention. - Please give at least one example of any object you define and at least one example of a property you define and one counter-example.
- Clear, formal statements of some propositions, lemmata, and theorems you found particularly interesting.
- Again, examples would be nice of these results
- The above definitions, results, and examples can all be mixed together to make some nice flow of material. The have a concluding section in which you mention something nice which can be done with the biggest, most interesting result you described.
- Please have a references section in which you give complete
references to the materials you used: books, articles, web
pages. Use whichever reference format you like, simply be
consistent and complete. Remember, the goal of a reference in
a scholarly paper is to enable readers to judge something
about the resource itself — such as: is it recent, was it
in a good journal,
*etc.*— and to find it themselves if they went to a library or hunted around on the Internet. (Note: Google Scholar is your friend in finding reference information!)

**LaTeX**) is encouraged but not required. Do make a draft and then a final version, that helps enormously in the quality of written work. - Students will give a short presentation on their project to the
class. You may project slides (
**LaTeX**or**Powerpoint**) if you like, or your could simply hand out a copy of the definitions and results part of your paper. In any case, you probably don't want to write out all the details of every definition and result on the board (although some writing on the board is fine). Mostly, you will be talking about the ideas, pointing to the details of the definitions and results on your slides or handout, and giving motivation.

- OK, it's the Hawaiian

Jonathan Poritz (jonathan.poritz@gmail.com) |
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