## Math 321 General Topology, 2018 Fall

### Instructor: Professor Jae Choon Cha

Office hour: by an appointment
Instructor’s web page: http://gt.postech.ac.kr/~jccha/

http://gt.postech.ac.kr/~jccha/general-topology-2018-fall/

### Classroom and hour

Monday and Wednesday 11:00–12:15pm, Math Building 402

### Syllabus

A brief version is available from POVIS. A more detailed pdf version is available here.

### Announcement: 1st Exam

November 8, Thursday, 8:00pm
Math Building 206

### Homework Problems

Homework problems will be posted on this web page, and will be collected in class on the due date. Turn it in BEFORE the lecture starts.  $\def\C{\mathbb{C}}\def\R{\mathbb{R}}\def\Q{\mathbb{Q}}\def\Z{\mathbb{Z}}\def\T{\mathcal{T}}\def\B{\mathcal{B}}\def\inte{\mathop{\text{int}}}$

Problem Set #5 (Due 11/19)

1. Show that any connected open set in $X$ is path connected if $X$ is locally path connected.
2. Suppose $p\colon X\to Y$ is a quotient map. Show that $Y$ is locally connected if $X$ is locally connected.
3. Let $X=S^1\times[-1,1]/\mathord{\sim}$, where $(z,t)\sim(w,s)$ if and only if either $(z,t)=(w,s)$ or $t=s=1$ or $t=s=-1$. Show that $X$ is homeomorphic to $S^2=\{x\in \R^3\mid \|x\|=1\}$ by constructing a homeomorphism. Hint: use compactness to show that it is a homeomorphism.
4. Show that the union of finitely many compact subsets of a space is compact.
5. Prove or disprove the following.
1. A compact subset of a metric space is bounded.
2. A compact subset of a metric space is closed.
3. A closed bounded subset of a metric space is compact.
6. For a function $f\colon X\to Y$, define the graph to be $G_f := \{(x,f(x))\in X\times Y\mid x\in X\}$. Suppose $Y$ is compact and Hausdorff. Show that $f$ is continuous if and only if $G_f$ is closed in $X\times Y$.
7. Suppose $\{A_n\}_{n=1}^\infty$ is a sequence of closed subsets in a compact Hausdorff space $X$. Show that $\inte \bigcup_n A_n=\emptyset$ if $\inte A_n=\emptyset$ for each $n$.
8. Suppose $(X,d)$ is a compact metric space. Let $\mathcal C$ be the collection of closed subsets of $X$. For $x\in X$ and $A\in \mathcal C$, define $d(x,A)=\inf\{d(x,a)\mid a\in A\}$.
1. Show that, for $x\in X$ and $A\in \mathcal C$, $d(x,A)=0$ if and only if $x\in A$.
2. Define $\rho(A,B)=\max\big\{\sup\{d(a,B)\mid a\in A\}, \sup\{d(b,A)\mid b\in B\}\big\}$ for $A,B\in \mathcal C$. Show that $\rho$ is a well-defined metric on $\mathcal C$.
3. Show that $(\mathcal C,\rho)$ is compact.
9. Suppose $X$ is limit point compact.
1. If $f\colon X\to Y$ is continuous, is $f(X)$ always limit point compact?
2. If $A$ is a closed subset of $X$, is $A$ always limit point compact?
3. If $X$ is a subspace of a Hausdorff space $Z$, is $X$ always closed in $Z$?
10. Suppose $(X,d)$ is a metric space. If $f\colon X\to X$ satisfies $d(x,y)=d(f(x),f(y))$ for all $x,y\in X$, then $f$ is called an isometry.
1. Show that an isometry is injective and continuous.
2. Show that an isometry on a compact space is a homeomorphism. Hint: to show the surjectivity, proceed as follows. If $x\not\in f(X)$, there is $\epsilon>0$ such that $B(x,\epsilon)\cap f(X)=\emptyset$; then for the sequence $\{x_n\}$ given by $x_1=x$ and $x_{n+1}=f(x_n)$, $d(x_n,x_m)\ge \epsilon$ whenever $n\ne m$.

Problem Set #4 (Due 11/07)

Read Munkres p.147-162. This week, I offer an opportunity to resubmit your solutions of #8, #9, #10 in the Problem Set #3. If you want, you may hand in an improved(hopefully!) solution of these problems as a part of this homework. Of course, it is not mandatory, and you don’t have to do it if you are already satisfied with your solutions submitted last time.

1. Give a detailed proof that $f(X)$ is path connected if $X$ is path connected and $f\colon X\to Y$ is continuous.
2. Suppose $A_1,A_2,\ldots$ are connected subspaces of a space $X$ such that $A_n\cap A_{n+1}\ne \emptyset$ for each $n$. Show that $\bigcup A_n$ is connected.
3. Suppose $\{A_\alpha\}_{\alpha\in J}$ is a collection of connected subspaces of a space $X$ and there is $\alpha_0\in J$ such that $A_{\alpha_0}\cap A_\alpha \ne \emptyset$ for each $\alpha\in J$. Show that $\bigcup A_\alpha$ is connected.
4. Suppose $A$ is a subset of a space $X$ and $C$ is a connected subspace of $X$ such that $C\cap A\ne \emptyset$, $C\cap(X-A)\ne\emptyset$. Show that $C$ intersects the boundary of $A$.
5. Suppose $X$ and $Y$ are connected spaces, and $A\subset X$ and $B\subset Y$ are proper subsets (i.e. their complements are nonempty). Show that $(X\times Y)-(A\times B)$ is connected.
6. Suppose $X_\alpha$ is connected for each $\alpha\in J$, where the index set $J$ is not necessarily finite. Show that the product space $X=\prod_{\alpha\in J} X_\alpha$ is connected, along the following outline.
1. Fix $a=(a_\alpha)\in X$. Let $\mathcal F$ be the collection of finite subsets of $J$. For each $K\in \mathcal F$, let $X_K = \{x =(x_\alpha)\in X\mid x_\alpha=a_\alpha$ for $\alpha\not\in K\}$. Show that $X_K$ is connected.
2. Show that $\bigcup_{K\in \mathcal F} X_K$ is connected.
3. Show that $X = \overline{\bigcup_{K\in \mathcal F} X_K}$, and from this conclude that $X$ is connected.
7. Give a detailed proof that the closure of $S=\{(x, \sin\frac1x)\mid x>0\}\subset \R^2$  is $\{0\}\times[-1,1] \cup S$.
8. Give a detailed proof that $S^n = \{x\in \R^{n+1} \mid |x|=1\}$ is path connected for $n\ge 1$.
9. Suppose $f\colon S^1\to \R$ is continuous. Show that there is $x\in S^1$ satisfying $f(x)=f(-x)$.
10. Give a detailed proof that $T = ([-1,1]\times\{0\})\cup (\{0\}\times [-1,0])\subset \R^2$ is not homeomorphic to $[0,1]$.
11. Show that $X\times Y$ is path connected if and only if $X$ and $Y$ are path connected.
12. Suppose $A$ is a countable subset of $\R^2$. Show that $\R^2-A$ is path connected.
13. Show that every open connected subset of $\R^n$ is path connected.
14. Identify the set $M_n(\R)$ of $n\times n$ real matrices with $\R^{n^2}$, under the correspondence $(a_{ij})_{1\le i,j\le n} \leftrightarrow (x_i)_{1\le i \le n^2}$ where $a_{ij}=x_{n(i-1)+j}$. Let $GL(n)=\{A\in M_n(\R)\mid A$ is nonsingular$\}$, $GL^+(n)=\{A\in GL(n)\mid \det A>0\}$.
1. Show that $GL(n)$ is not connected using the continuity of determinant.
2. Show that $GL^+(n)$ is path connected. (Hint: you may find a path from a given $A\in GL^+(n)$ to $I$ by decomposing $A$ into elementary matrices.)
3. How many path components does $GL(n)$ have? Give a proof.

Problem Set #3 (Due 10/26)

Read Munkres p. 119-144 (you may skip Example 1 on page 132)

1. Give detailed proofs of the following statements:
1. For every metric $d$, the standard bounded metric $\bar d(x,y)=\min\{d(x,y),1\}$ is a metric.
2. Let $f(t)=\frac{t}{1+t}$. Then for any metric $d$, $\rho(x,y)=f(d(x,y))$ is a bounded metric.
2. Recall $\R^\omega = \R\times \R\times \cdots = \prod_{i=1}^\infty \R$. Let $\R^\infty =\{x=(x_n)\in \R^\omega \mid x_n = 0$ for all $n$ but finitely many$\}$. Determine the closure of $\R^\infty$ in $\R^\omega$ with a proof.
3. When $\R^\omega$ is equipped with the uniform metric of the standard bounded metric on $\R$, determine the closure of $\R^\infty$ in $\R^\omega$ with a proof.
4. On $\R^n$, define $D(x,y)=d(x_1,y_1)+ \cdots + d(x_n,y_n)$. Show that $D$ is a metric which induces the standard topology on $\R^n$.
5. Show that the dictionary topology on $\R\times \R$ is metrizable.
6. Let $X$ be a topological space and $(Y,d)$ be a metric space. We say that a sequence $\{f_n\}$ of functions $f_n\colon X\to Y$ uniformly converges to a function $f\colon X\to Y$ if for every $\epsilon>0$, there is $N>0$ such that $d(f_n(x),f(x)) < \epsilon$ for all $n>N$ and all $x\in X$. Prove that if each $f_n$ is continuous and if $\{f_n\}$ uniformly converges to $f$, then $f$ is continuous.
7. Suppose $Y$ is a set. Let $X_y = \R$ for each $y\in Y$. Denote the product space $\prod_{y\in Y} X_y$ by $\R^Y$. In other words, $\R^Y = \{x=(x_y)_{y\in Y} \mid x_y\in \R\}$. Recall that the uniform metric $\rho$ on $\R^Y$ is defined from the standard bounded metric on $\R$. On the other hand, $\R^Y$ can be identified with the set of functions $Y\to \R$. Show that a sequence $\{f_n\}$ of functions $f_n\colon Y\to \R$ converges uniformly to a function $f\colon Y\to \R$ if and only if $\{f_n\}$ converges to $f$ in the metric space $(\R^Y,\rho)$.
8. Show the following:
1. Suppose $p\colon X \to Y$ is a continuous map which has a continuous left inverse, that is, there is a continuous map $q\colon Y\to X$ satisfying $p\circ q = 1_Y$ (identity map on $Y$). Then $p$ is a quotient map.
2. retraction of a space $X$ onto its subspace $A$ is a continuous map $r\colon X\to A$ satisfying that $r(a)=a$ for every $a\in A$.  Show that a retraction is a quotient map.
9. Define an equivalence relation $\sim$ on $\R^2$ by $(x,y)\sim(z,w)$ if and only if $x^2+y^2 = z^2+w^2$. Our goal is to show that the quotient space $\R^2/{\sim}$ is homeomorphic to the ray $[0,\infty)=\{x\in \R \mid x\ge 0\}$. Let $p\colon \R^2 \to \R^2/{\sim}$ be the projection.
1. Define $g\colon \R^2\to [0,\infty)$ by $g(x,y)=\sqrt{x^2+y^2}$. Show that $g$ induces a continuous bijection $f\colon\R^2/\mathord{\sim} \to [0,\infty)$.
2. Find a continuous map $h\colon [0,\infty) \to \R^2$ such that $f\circ p\circ h = 1_{[0,\infty)}$ (identity map on $[0,\infty)$). Conclude that $f$ is a homeomorphism.
10. Let $X^*$ be the quotient space $S^1\times S^1/{\sim}$, where $(z,w)\sim (u,v)$ if and only if $zw=uv$. Here $S^1=\{z\in \C\mid |z|=1\}$ as we defined in class. Show that $X^*$ is homeomorphic to $S^1$. (Your solution to #9 may be helpful!)

Problem Set #2 (Due 10/17)

1. Suppose $Y$ is a subspace of a space $X$, and let $i\colon Y \to X$ be the inclusion $i(y)=y$.
1. Show that the subspace topology on $Y$ is the smallest topology for which $i$ is continuous.
2. Show that if $f\colon X\to Z$ is continuous, then its restriction $f\mid_Y \colon Y\to Z$ is continuous.
3. Show that a function $f\colon Z\to Y$ of a space $Z$ is continuous if and only if the composition $i\circ f$ is continuous.
2. Suppose $X$ is a topological space, and $f,\;g\colon X\to \R$ are two real-valued continuous functions. Define $f+g\colon X\to \R$ by $(f+g)(x)=f(x)+g(x)$, and define $f-g$, $f\cdot g\colon X\to \R$ similarly. Show that $f+g$, $f-g$, $f\cdot g$ are continuous.  When $g(x)\ne 0$ for all $x\in X$, show that $f/g\colon X\to \R$ defined by $(f/g)(x)=f(x)/g(x)$ is continuous.
3. Let $X = \{(x,y)\in \R^2\mid x^2+y^2\le 1\}$ and $Y = \{(x,y)\in \R^2\mid x^2+y^2\le 1, x\ge 0\}$ be subspaces of $\R^2$. Show that $X$ and $Y$ are homeomorphic (you may use the above #1, #2).
4. Show that if $A\subset X$ and $B\subset Y$ are closed sets of spaces $X$ and $Y$, then $A\times B$ is closed in $X\times Y$.
5. Prove the following.
1. $\overline{A\cup B} = \overline A \cup \overline B$.
2. $\bigcup \overline{A_\alpha} \subset \overline{\bigcup A_\alpha}$.
6. Disprove the following by giving a counterexample.
1. $\bigcup \overline{A_\alpha} = \overline{\bigcup A_\alpha}$.
2. $\overline{A\cap B} = \overline A \cap \overline B$.
3. $\overline {A -B} = {\overline A} – {\overline B}$.
7. Show that the following spaces are Hausdorff.
1. A simply ordered set with the order topology.
2. A subspace of a Hausdorff space.
3. The product of two Hausdorff spaces.
8. Show that a space $X$ is Hausdorff if and only if the subset $D=\{(x,x)\mid x\in X\}$ is closed in $X\times X$. (The subset $D$ is called the diagonal.)
9. Denote by $\partial A$ the boundary of a subset $A$ of a space $X$. Show the following.
1. $\inte A \cap \partial A = \emptyset$, and ${\overline A} = \inte A \cup \partial A$.
2. $\partial A$ is empty if and only if $A$ is both open and closed.
3. $A$ is open if and only if $\partial A = {\overline A} – A$.
10. Suppose $f,g\colon X\to \R$ are continuous.
1. Show that the subset $\{x\in X \mid f(x)\le g(x)\}$ is closed in $X$.
2. Show that $h(x)=\max\{f(x), g(x)\}$ is continuous on $X$.
11. Give a detailed proof that our basis for the product topology on $\prod_{\alpha} X_\alpha$  defined in class is indeed a basis.
12. Suppose $\{x_1,x_2,\ldots\}$ is a sequence in the product space $\prod_\alpha X_\alpha$.
1. Show that the sequence converges to $x\in \prod_\alpha X_\alpha$ if and only if the sequence $\{\pi_\alpha(x_1),\pi_\alpha(x_2),\ldots\}$ converges to $\pi_\alpha(x)$ for each index $\alpha$.
2. Is the statement in (a) still true when the box topology is used?

Problem Set #1 (Due 09/27 23:59pm)

Read Munkres p.75-88 (you may skip Lemma 13.4), p.102-108 (not including Theorem 18.1 and 18.3 for now).

1. Suppose $X$ is a set.
1. Let $\T$ be the collection of subsets $U$ of $X$ such that $X-U$ is either finite or equal to $X$. Show that $\T$ is a topology on $X$.  ($\T$ is called the finite complement topology.)
2. Let $\T$ be the collection of subsets $U$ of $X$ such that $X-U$ is either countable or equal to $X$. Show that $\T$ is a topology on $X$.
3. Let $\T$ be the collection of subsets $U$ of $X$ such that $X-U$ is either infinite, empty or equal to $X$. Is $\T$ a topology? Prove or disprove by a counterexample.
2. Let $\B=\{(a,b) \subset \R \mid a,b\in \Q,\; a< b\}$ and $\B’=\{(a/2^n,b/2^n) \subset \R \mid a,b,n\in \Z,\; a< b\}$.
1. Show that both $\B$ and $\B’$ are bases on $\R$.
2. Show that $\B$ and $\B’$ generate the same topology on $\R$.
3. Suppose $\{\T_\alpha\}$ is a collection of topologies $\T_\alpha$ on a fixed set $X$.
1. Show that $\bigcap_\alpha \T_\alpha$ is a topology on $X$.
2. Is $\bigcup_\alpha \T_\alpha$ a topology? Prove or disprove by a counterexample.
4. For two topologies $\T$ and $\T’$ on a set $X$, if $\T \subset \T’$, then we say that $\T$ is smaller than $\T’$ (and $\T’$ is larger than $\T$; often we also say $\T$ is coarser than $\T’$ and $\T’$ is finer than $\T$).
1. What is the largest topology among all the topologies on a fixed set $X$?
2. Suppose $\{\T_\alpha\}$ is a collection of topologies on a set $X$. Show that there is a unique largest topology contained in every $\T_\alpha$.
3. Suppose $\B$ is a base for a topology on $X$. Show that the topology $\T$ generated by $\B$ is the smallest topology among the topologies on $X$ containing $\B$.  (In order words, $\T$ contains $\B$, and if $\T’$ is a topology containing $\B$, then $\T$ is smaller than $\T’$.)
5. Suppose $X$ and $Y$ are spaces and $y_0\in Y$. Let $c\colon X\to Y$ be the constant function defined by $c(x)=y_0$. Show that $c$ is continuous.
6. Give a rigorous proof that any two (nonempty) open intervals $(a,b)$ and $(c,d)$ in $\R$ are homeomorphic.  Here $(a,b)$ and $(c,d)$ are equipped with the order topology.
7. The dictionary order on $\R\times\R$ is defined as follows: $(x,y)<(x’,y’)$ if and only if either $x<x’$, or $x=x’$ and $y<y’$.  Let $\R_d$ be the set $\R$ equipped with the discrete topology. Show that the product topology on $\R_d\times \R$ is equal to the dictionary order topology on $\R\times \R$.