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/

Course home page

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)

Read Munkres 163-182.

  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)

Read Munkres p.88-117.

  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 $\R\times \R$.