Homework Assignments
Homework is collected at the beginning of lecture of the specified day. No late homework will be accepted. Please write your name and the homework number on each assignment. If you would like to try writing your homework with LaTeX, here is a template that produces this output.
Homework 12, due Friday, December 4th ($\S 27, 28, 31, 33$) Solutions
- 1. Let $X$ be a compact topological space and $(Y,d)$ a metric space. Let $C(X,Y)$ denote the set of all continuous functions $f\colon X\to Y$.
- (a) For $f,g\in C(X,Y)$, show that $h\colon X\to \mathbb{R}$ defined by $h(x):=d(f(x),g(x))$ is continuous.
- (b) Show that $D(f,g):=\sup_{x\in X} d(f(x), g(x))$ exists and defines a metric on $C(X,Y)$.
- (c) Let $\varphi\colon X\to X$ be a continuous function. Show that the map $\Phi\colon C(X,Y)\to C(X,Y)$ defined by $\Phi(f):=f\circ\varphi$ is uniformly continuous with respect to the metric $D$.
- 2. Let $(X,d)$ be a metric space. For $x\in X$ and nonempty $A\subset X$, recall that $d(x,A):=\inf_{a\in A} d(x,a)$.
- (a) Show that $d(x,A)=0$ if and only if $x\in \overline{A}$.
- (b) Suppose $A\subset V$ for $A$ compact and $V$ open. Show that there exists $\epsilon>0$ so that \[ \bigcup_{a\in A} B_d(a,\epsilon)\subset V. \] [Hint: consider the function $f(x)=d(x,X\setminus V)$.]
- 3. Let $(X,d)$ be a compact metric space and let $f\colon X\to X$ be a function satisfying $d(f(x),f(y)) = d(x,y)$ for all $x,y\in X$. (We call such a function an isometry.) Show that $f$ is a homeomorphism.
- 4. Let $X$ be a normal topological space and let $A,B\subset X$ be disjoint closed subsets of $X$. Show that there are open subsets $U,V\subset X$ satisfying $A\subset U$, $B\subset V$, and $\overline{U}\cap \overline{V}=\emptyset$.
- 5. Let $X$ be a normal topological space. We say $A\subset X$ is a $G_\delta$ set if it is a countable intersection of open sets. Show that $A\subset X$ is a closed $G_\delta$ set if and only if there exists a continuous function $f\colon X\to [0,1]$ with $f(x)=0$ for all $x\in A$ and $f(x)>0$ for all $x\not\in A$. [Hint: use Urysohn's Lemma.]
- 6*. For $d\in \mathbb{N}$ and $a=0,1,\ldots, d-1$ define
\[
U_{d,a}:=\{dn+a\mid n\in \mathbb{Z}\} \subset \mathbb{Z}.
\]
In this exercise you will use topology to show that there are infinitely many prime numbers.
- (a) Show that the collection $\mathcal{B}:=\{U_{d,a}\mid d\in \mathbb{N},\ a=0,1,\ldots, d-1\}$ forms a basis for a topology on $\mathbb{Z}$.
- (b) Show that $U_{d,a}$ is clopen in this topology.
- (c) Show that if $U\subset \mathbb{Z}$ is nonempty and open in this topology, then $U$ is infinite.
- (d) Let $P\subset \mathbb{N}$ be the subset of prime numbers. Consider \[ A:=\bigcup_{p\in P} U_{p,0}. \] Show that $\mathbb{Z}\setminus A$ is finite.
- (e) Deduce that $P$ is infinite.
Homework 11, due Friday, November 20th ($\S 24, 26$) Solutions
- 1. Recall that $S^1=\{(x,y)\in \mathbb{R}^2\mid x^2+y^2=1\}$.
- (a) Show that $S^1$ is connected.
- (b) Show that $a(x,y):=(-x,-y)$ defines a homeomorphism $a\colon S^1\to S^1$.
- (c) Show that if $f\colon S^1\to\mathbb{R}$ is continuous, then there exists $(x,y)\in S^1$ satisfying $f(x,y)= f(-x,-y)$.
- 2. Let $U\subset \mathbb{R}^n$ be open and connected. Show that $U$ is path connected. [Hint: for $\mathbf{x}_0\in U$ show that the set of points $\mathbf{x}\in U$ that are connected to $\mathbf{x}_0$ by a path in $U$ is clopen.]
- 3. Equip $\mathbb{R}$ with the finite complement topology. Show that every subset is compact.
- 4. Let $X$ be a Hausdorff space. If $A,B\subset X$ are compact with $A\cap B=\emptyset$, show that there are open sets $U\supset A$ and $V\supset B$ with $U\cap V=\emptyset$.
- 5. Let $p\colon X\to Y$ be a closed continuous surjective map.
- (a) For $U\subset X$ open, show that $p^{-1}(\{y\})\subset U$ for $y\in Y$ implies there is a neighborhood $V$ of $y$ with $p^{-1}(V)\subset U$.
- (b) Show that if $Y$ is compact and $p^{-1}(\{y\})$ is compact for each $y\in Y$, then $X$ is compact.
- 6*. Let $G$ be a topological group with identity $e\in G$.
- (a) For $U\subset G$ a neighborhood of $e$, show that there exists a neighborhood $V$ of $e$ satisfying $VV\subset U$.
- (b) For $A\subset G$ closed and $B\subset G$ compact with $A\cap B=\emptyset$, show that there exists a neighborhood $V$ of $e$ satisfying $A\cap VB=\emptyset$.
- (c) For $A\subset G$ closed and $B\subset G$ compact, show that $AB$ is closed.
- (d) For $H < G$ a compact subgroup, show that the quotient map $p\colon G\to G/H$ is closed.
- (e) Show that if $H < G$ is a compact subgroup with $G/H$ compact, then $G$ is compact.
Homework 10, due Friday, November 13th ($\S 22, 23$) Solutions
- 1. Recall that for $\mathbf{x}=(x_1,x_2)\in\mathbb{R}^2$, its norm is $\|\mathbf{x}\|=(x_1^2+x_2^2)^{1/2}$. Consider $X:=\mathbf{R}^2\setminus\{(0,0)\}$ and $S^1:=\{\mathbf{x}\in\mathbb{R}^2\mid \|\mathbf{x}\|=1\}$ equipped with their subspace topologies, where $\mathbb{R}^2$ has its standard topology.
- (a) Show that $p(\mathbf{x}):=\frac{1}{\|\mathbf{x}\|}\mathbf{x}$ defines a continuous map $p\colon X\to S^1$.
- (b) Show that $p$ is a quotient map.
- (c) Define an equivalence relation $\sim$ on $X$ so that the quotient space $X/\sim$ is homeomorphic to $S^1$. Give a geometric description of the equivalence classes.
- 2. Prove whether or each of the following spaces is connected or disconnected.
- (a) $\mathbb{R}$ equipped with the lower limit topology.
- (b) $\mathbb{R}$ equipped with the finite complement topology.
- (c) $\mathbb{R}^\mathbb{N}$ equipped with the uniform topology.
- 3. Let $X$ be a topological space and let $\{Y_j\mid j\in J\}$ be an indexed family of connected subspaces of $X$. Suppose there exists a connected subspace $Y\subset X$ satisfying $Y\cap Y_j\neq\emptyset$ for all $j\in J$. Show that $Y\cup \bigcup_{j\in J} Y_j$ is connected.
- 4. Let $X$ and $Y$ be connected spaces and let $A\subsetneq X$ and $B\subsetneq Y$ be proper subsets. Show that $(X\times Y)\setminus (A\times B)$ is connected.
- 5. Let $p\colon X\to Y $ be a quotient map. Suppose that $Y$ is connected and $p^{-1}(\{y\})$ is connected for every $y\in Y$. Show that $X$ is connected.
- 6*. Let $C_0:=[0,1]\subset\mathbb{R}$ and for each $n\in \mathbb{N}$ recursively define
\[
C_n := C_{n-1}\setminus \bigcup_{k=0}^{3^{n-1}-1} \left(\frac{1+3k}{3^n},\frac{2+3k}{3^n}\right).
\]
Then $C:=\bigcap_{n=0}^\infty C_n$ is called the Cantor set. Equip $C\subset \mathbb{R}$ with the subspace topology.
- (a) Show $C=\overline{C}\setminus C^\circ$.
- (b) Show that every $x\in C$ is a limit point of $C$.
- (c) Show that $C$ is totally disconnected: singleton sets are the only connected subsets.
Homework 9, due Friday, November 6th ($\S 21, 22$) Solutions
- 1. Let $X$ be a set and let $Y$ be a metric space with metric $d$. Define a metric on $Y^X$ by
\[
\overline{\rho}( (y_x)_{x\in X}, (z_x)_{x\in X}) := \sup_{x\in X} \overline{d}(y_x,z_x),
\]
where $\overline{d}(y,z) = \min\{ d(y,z),1\}$ is the standard bounded metric corresponding to $d$. Let $f_n,f\colon X\to Y$ be functions, $n\in \mathbb{N}$, and define $\mathbf{f_n}, \mathbf{f}\in Y^X$ by $\mathbf{f_n}:=(f_n(x))_{x\in X}$ and $\mathbf{f}:=(f(x))_{x\in X}$.
- (a) Show that $(f_n)_{n\in \mathbb{N}}$ converges pointwise to $f$ if and only if the sequence $( \mathbf{f_n} )_{n\in \mathbb{N}}$ converges to $\mathbf{f}$ when $Y^X$ is given the product topology.
- (b) Show that $(f_n)_{n\in \mathbb{N}}$ converges uniformly to $f$ if and only if the sequence $( \mathbf{f_n} )_{n\in \mathbb{N}}$ converges to $\mathbf{f}$ when $Y^X$ is given the topology induced by the metric $\overline{\rho}$.
- 2. Let $X$ be a topological space. For a subset $A\subset X$, a retraction of $X$ onto $A$ is a continuous map $r\colon X\to A$ satisfying $r(a)=a$ for all $a\in A$.
- (a) Let $p\colon X\to Y$ be a continuous map between topological spaces. Show that if there exists a continuous function $f\colon Y\to X$ so that $p(f(y))=y$ for all $y\in Y$, then $p$ is a quotient map.
- (b) Show that a retraction is a quotient map.
- 3. Consider the following subset of $\mathbb{R}^2$: \[ A:=\{(x,y)\in \mathbb{R}^2\mid \text{either }x\geq 0 \text{ or }y=0 \text{ (or both)}\}. \] Define $q\colon A\to \mathbb{R}$ by $q(x,y)=x$. Show that $q$ is a quotient map, but is neither open nor closed.
- 4. Let $X$ and $Y$ be topological spaces and let $p\colon X\to Y$ be a surjective map.
- (a) Show that a subset $A\subset X$ is saturated with respect to $p$ if and only if $X\setminus A$ is saturated with respect to $p$.
- (b) Show that $p(U)\subset Y$ is open for all saturated open sets $U\subset X$ if and only if $p(A)\subset Y$ is closed for all saturated closed sets $A\subset X$.
- (c) Show that if $p$ is an injective quotient map, then it is a homeomorphism.
- 5. Let $X:=(0,1]\cup [2,3)$, $Y:=(0,2)$, and $Z:=(0,1]\cup (2,3)$ and define maps $p\colon X\to Y$ and $q\colon X\to Z$ by
\[
p(t):=\begin{cases} t & \text{if }0 < t\leq 1 \\ t-1 & \text{if }2\leq t < 3\end{cases} \qquad\text{ and }\qquad q(t):=\begin{cases} t &\text{if } t\neq 2 \\ 1 & \text{otherwise} \end{cases}.
\]
Equip $X$ and $Y$ with their subspace topologies from $\mathbb{R}$ and equip $Z$ with the quotient topology induced by $q$.
- (a) Show that $p$ is a quotient map.
- (b) Show that $q$ is a quotient map.
- (c) Show that $f\colon Y\to Z$ defined by \[ f(t):=\begin{cases} t & \text{if }0 < t\leq 1 \\ t+1 & \text{if } 1 < t < 2 \end{cases} \] is a homeomorphism. [Hint: show $f\circ p = q$.]
- 6*. Consider
\begin{align*}
X&:=\{\mathbf{x}\in \mathbb{R}^2\mid \|\mathbf{x}\|\leq 1\}\\
S^2&:=\{\mathbf{x}\in \mathbb{R}^3\mid \|\mathbf{x}\|=1\}.
\end{align*}
In this exercise you will show a quotient space of $X$ is homeomorphic to $S^2$.
- (a) Let $S^1:=\{\mathbf{x}\in \mathbb{R}^2\mid \|\mathbf{x}\|=1\}$. Show that $f\colon X\setminus S^1\to \mathbb{R}^2$ defined by \[ f(\mathbf{x}) := \frac{1}{1-\|\mathbf{x}\|}\mathbf{x} \] is a homeomorphism.
- (b) Show that $g\colon S^2\setminus\{(0,0,1)\}\to \mathbb{R}^2$ defined by \[ g(\mathbf{x}):= \frac{1}{1-x_3}(x_1,x_2) \] is a homeomorphism.
- (c) Show that $p\colon X\to S^2$ defined by \[ p(\mathbf{x}) := \begin{cases} g^{-1}\circ f(\mathbf{x}) & \text{if } \mathbf{x}\in X\setminus S^1 \\ (0,0,1) & \text{otherwise} \end{cases} \] is a quotient map.
- (d) Define an equivalence relation on $X$ by $\mathbf{x}\sim\mathbf{y}$ if and only if $p(\mathbf{x})=p(\mathbf{y})$. Describe the quotient space $X/\sim$ and show that it is homeomorphic to $S^2$.
Homework 8, due Friday, October 30th ($\S 20, 21$) Solutions
- 1. Let $X$ be a metric space with metric $d$. Prove the reverse triangle inequality: for all $x,y,z\in X$ \[ | d(x,y) - d(y,z) |\leq d(x,z). \]
- 2. Recall that the uniform metric on $\mathbb{R}^\mathbb{N}$ is defined as
\[
\overline{\rho}(\mathbf{x},\mathbf{y}) = \sup_{n\in \mathbb{N}} \left(\min\{|x_n-y_n|,1\}\right).
\]
- (a) Show that $\overline{\rho}$ is a metric.
- (b) Let $C\subset \mathbb{R}^\mathbb{N}$ be the subset from Exercise 4 on Homework 6. Determine $\overline{C}$ when $\mathbb{R}^\mathbb{N}$ has the topology induced by $\overline{\rho}$.
- (c) Let $h\colon \mathbb{R}^\mathbb{N}\to\mathbb{R}^\mathbb{N}$ be the function from Exercise 1 on Homework 7. Find necessary and sufficient conditions on the sequences $(a_n)_{n\in \mathbb{N}}, (b_n)_{n\in \mathbb{N}}$ which guarantee $h$ is continuous when $\mathbb{R}^\mathbb{N}$ has the topology induced by $\overline{\rho}$.
- (d) For $\mathbf{x}\in \mathbb{R}^\mathbb{N}$ and $\epsilon>0$, show that \[ U:=(x_1-\epsilon, x_1+\epsilon)\times (x_2-\epsilon, x_2+\epsilon) \times \cdots \] is not open with respect to the topology induced by $\overline{\rho}$.
- 3. Let $X$ be a metric space with metric $d$. For fixed $x_0\in X$, show that the function $f\colon X\to \mathbb{R}$ defined by $f(x)=d(x,x_0)$ is continuous.
- 4. Let $X$ be a metric space with metric $d$, and let $(x_i)_{i\in I}\subset X$ be a net.
- (a) Show that $(x_i)_{i\in I}$ converges to $x_0\in X$ if and only if the net $( d(x_i, x_0) )_{i\in I}\subset \mathbb{R}$ converges to $0$.
- (b) Show that if $(x_i)_{i\in I}$ converges to $x_0\in X$, then one can find a sequence $(x_n)_{n\in \mathbb{N}}\subset \{x_i\mid i\in I\}$ converging to $x_0$.
- 5. For each $n\in \mathbb{N}$, define $f_n\colon\mathbb{R}\to\mathbb{R}$ by \[ f_n(x) = \frac{1}{1+(x-n)^2}. \] Show that the sequence of functions $(f_n)_{n\in \mathbb{N}}$ converges to the zero function pointwise but not uniformly.
- 6*. Let $\ell^2\subset \mathbb{R}^\mathbb{N}$ be the set of sequences $(x_n)_{n\in \mathbb{N}}$ for which the series $\sum_{n=1}^\infty x_n^2$ converges. For $\mathbf{x}=(x_n)_{n\in\mathbb{N}}\in \ell^2$ denote
\[
\|\mathbf{x}\|_2:= \left( \sum_{n=1}^\infty x_n^2 \right)^{1/2}.
\]
- (a) For $\mathbf{x}\in \ell^2$ and $c\in \mathbb{R}$, show that $c\mathbf{x}\in \ell^2$ with $\|c\mathbf{x}\|_2 = |c| \|\mathbf{x}\|_2$.
- (b) For $\mathbf{x},\mathbf{y}\in \ell^2$, show that the series $\sum_{n=1}^\infty |x_ny_n|$ converges and is bounded by $\|\mathbf{x}\|_2\|\mathbf{y}\|_2$.
- (c) For $\mathbf{x},\mathbf{y}\in \ell^2$, show that $\mathbf{x}+\mathbf{y}\in \ell^2$ with $\|\mathbf{x}+\mathbf{y}\|_2\leq \|\mathbf{x}\|_2 + \|\mathbf{y}\|_2$.
- (d) Show that $d_2(\mathbf{x},\mathbf{y})=\|\mathbf{x} - \mathbf{y}\|_2$ defines a metric on $\ell^2$.
- (e) Show that the topology induced by $d_2$ is finer than the uniform topology but coarser than the box topology on $\ell^2$.
Homework 7, due Friday, October 23rd ($\S 19, 20$) Solutions
- 1. Let $(a_n)_{n\in \mathbb{N}},(b_n)_{\in \mathbb{N}}\in \mathbb{R}^\mathbb{N}$ with $a_n>0$ for all $n\in \mathbb{N}$. Define a map $h\colon \mathbb{R}^\mathbb{N}\to \mathbb{R}^\mathbb{N}$ by
\[
h( (x_n)_{n\in \mathbb{N}}) = ( a_n x_n + b_n )_{n\in\mathbb{N}}.
\]
- (a) Show that $h$ is a bijection.
- (b) Show that if $\mathbb{R}^\mathbb{N}$ is given the product topology, then $h$ is a homeomorphism.
- (c) Prove whether or not $h$ is a homeomorphism when $\mathbb{R}^\mathbb{N}$ is given the box topology.
- 2. For $\mathbf{x}=(x_1,\ldots, x_n), \mathbf{y}=(y_1,\ldots, y_n)\in\mathbb{R}^n$, define
\[
d_1(\mathbf{x},\mathbf{y}):= \sum_{j=1}^n |x_j - y_j|.
\]
- (a) Show that $d_1$ is a metric on $\mathbb{R}^n$.
- (b) Show that the topology induced by $d_1$ equals the product topology on $\mathbb{R}^n$.
- (c) For $n=2$ and $\mathbf{0}=(0,0)\in \mathbb{R}^2$, draw a picture of $B_{d_1}(\mathbf{0},1)$.
- 3. Let $X$ be a metric space with metric $d$. For $x\in X$ and $\epsilon>0$, show that $\{y\in X\mid d(x,y)\leq \epsilon\}$ is a closed set.
- 4. Let $X$ be a metric space with metric $d$. Show that $d\colon X\times X\to \mathbb{R}$ is continuous.
- 5. For $\mathbf{x}=(x_1,\ldots, x_n), \mathbf{y}=(y_1,\ldots, y_n)\in\mathbb{R}^n$ and $c\in \mathbb{R}$ define
\begin{align*}
\mathbf{x}+\mathbf{y}&:=(x_1+y_1,\ldots, x_n+y_n),\\
c\mathbf{x} &:= (c x_1,\ldots, cx_n),\\
\mathbf{x}\cdot\mathbf{y}&:=x_1y_1+\cdots +x_ny_n,\\
\|\mathbf{x}\|&:=(x_1^2+\cdots +x_n^2)^{1/2}.
\end{align*}
- (a) For $\mathbf{x},\mathbf{y},\mathbf{z}\in \mathbb{R}^n$ and $a,b\in \mathbb{R}$, prove the following formulas \begin{align*} \|\mathbf{x}\|^2 &= \mathbf{x}\cdot\mathbf{x}\\ (a\mathbf{x})\cdot (b\mathbf{y})&=(ab)(\mathbf{x}\cdot\mathbf{y})\\ \mathbf{x}\cdot\mathbf{y}&=\mathbf{y}\cdot\mathbf{x}\\ \mathbf{x}\cdot(\mathbf{y}+\mathbf{z}) &= \mathbf{x}\cdot \mathbf{y}+\mathbf{x}\cdot \mathbf{z} \end{align*}
- (b) Show that $|\mathbf{x}\cdot\mathbf{y}| \leq \|\mathbf{x}\| \|\mathbf{y}\|$.
[Hint: for $\mathbf{x},\mathbf{y}\neq 0$ let $a=\frac{1}{\|\mathbf{x}\|}$ and $b=\frac{1}{\|\mathbf{y}\|}$ and use the fact that $\| a\mathbf{x} \pm b\mathbf{y}\|^2\geq 0$.] - (c) Show that $\|\mathbf{x}+\mathbf{y}\| \leq \|\mathbf{x}\|+\|\mathbf{y}\|$.
- (d) Prove that the euclidean metric $d(\mathbf{x},\mathbf{y}):=\|\mathbf{x} - \mathbf{y}\|$ is indeed a metric.
- 6*. For $\mathbf{x}=(x_1,\ldots, x_n)\in\mathbb{R}^n$ and $1\leq p < \infty$, define
\[
\| \mathbf{x}\|_p:=(|x_1|^p+\cdots +|x_n|^p)^{1/p},
\]
and for $p=\infty$ define
\[
\|\mathbf{x}\|_\infty:= \max \{|x_1|, \ldots, |x_n|\}.
\]
In this exercise you will show $d_p(\mathbf{x},\mathbf{y}):=\| \mathbf{x} - \mathbf{y}\|_p$ defines a metric for each $1\leq p\leq \infty$. Observe that $p=1,2,\infty$ yield the metric from Exercise 2, the euclidean metric, and the square metric, respectively.
- (a) For $1 < p < \infty$, show that if $q > 0$ satisfies $\frac{1}{p}+\frac{1}{q}=1$ then $1 < q < \infty$. We call $q$ the conjugate exponent to $p$.
- (b) For $a,b\geq 0$ and $0< \lambda <1$, show that $a^\lambda b^{1-\lambda} \leq \lambda a + (1-\lambda) b$.
- (c) Prove Hölder's Inequality: for $1 < p < \infty$ with conjugate exponent $q$ and $\mathbf{x},\mathbf{y}\in \mathbb{R}^n$ show that \[ |x_1y_1|+\cdots +|x_ny_n| \leq \|\mathbf{x}\|_p \|\mathbf{y}\|_q. \]
- (d) Prove Minkowski's Inequality: for $1 < p < \infty$ and $\mathbf{x},\mathbf{y}\in \mathbb{R}^n$ show that \[ \|\mathbf{x}+\mathbf{y}\|_p \leq \|\mathbf{x}\|_p + \|\mathbf{y}\|_p. \] [Hint: use $|x_j+y_j|^p \leq (|x_j|+|y_j|)|x_j+y_j|^{p-1}$.]
- (e) Show that $d_p$ is a metric for $1 < p < \infty$.
- (f) Show that the topology induced by $d_p$ equals the product topology on $\mathbb{R}^n$ for $1 < p < \infty$, where $\mathbb{R}$ has the standard topology.
[Hint: show that $\|\mathbf{x}\|_\infty\leq \|\mathbf{x}\|_p\leq \|\mathbf{x}\|_1$.]
Homework 6, due Friday, October 16th ($\S 18, 19$) Solutions
- 1. Let $A,B,C,D$ be topological spaces and suppose $f\colon A\to B$ and $g\colon C\to D$ are continuous functions. Define a function $f\times g\colon A\times C\to B\times D$ by \[ (f\times g)(a,c) = ((f(a), g(c)). \] Show that $f\times g$ is continuous when $A\times C$ and $B\times D$ are given the product topologies.
- 2. Let $\mathbb{R}$ and $\mathbb{R}^2$ have their standard topologies.
- (a) Show that the function $f\colon \mathbb{R}^2\to\mathbb{R}$ defined by $f(x,y)=xy$ is continuous.
- (b) For each $n\in \mathbb{N}$, show that $p\colon \mathbb{R}\to \mathbb{R}$ defined by $p(x)=x^n$ is continuous.
- 3. Let $X$ be a topological space and let $Y$ be set with order relation $<$ and the order topology. Suppose $f,g\colon X\to Y$ are continuous.
- (a) Show that the set $\{x\in X \mid f(x)\leq g(x)\}$ is closed in $X$.
- (b) Show that the function $h\colon X\to Y$ defined by $h(x):=\min\{f(x),g(x)\}$ is continuous. [Hint: using the pasting lemma.]
- 4. Let $\mathbb{R}$ have the standard topology. Consider
\[
C = \{ (x_n)_{n\in \mathbb{N}} \in \mathbb{R}^\mathbb{N}\mid x_n\neq 0 \text{ for only finitely many }n\in\mathbb{N}\}.
\]
That is, $C$ is the set of sequences that are eventually equal to zero.
- (a) Determine $\overline{C}$ when $\mathbb{R}^\mathbb{N}$ has the box topology.
- (b) Determine $\overline{C}$ when $\mathbb{R}^\mathbb{N}$ has the product topology.
- 5. Let $\{X_j\mid j\in J\}$ be an indexed family of topological spaces. Let $(\mathbf{x}_i)_{i\in I} \subset \prod_{j\in J} X_j$ be a net; that is, for each $i$ in the directed set $I$, $\mathbf{x}_i\in \prod_{j\in J} X_j$ is a $J$-tuple.
- (a) Equip $\prod_{j\in J} X_j$ with the product topology and show that the net $(\mathbf{x}_i)_{i\in I}$ converges to some $\mathbf{x}\in \prod_{j\in J} X_j$ if and only if for every $j\in J$ the net $(\pi_j(\mathbf{x}_i))_{i\in I}$ converges to $\pi_j(\mathbf{x})$ in $X_j$.
- (b) Equip $\prod_{j\in J} X_j$ with the box topology and prove one of the directions in the previous part is true and show the other is false by finding a counterexample in $\mathbb{R}^\mathbb{N}$.
- 6*. Let $\mathbb{R}$ have the standard topology and consider the functions $f,g\colon \mathbb{R}\to\mathbb{R}$ defined by
\[
f(x)= \begin{cases} 1 & x\in \mathbb{Q}\\
0 & x\in \mathbb{R}\setminus\mathbb{Q} \end{cases},
\]
and
\[
g(x)=\begin{cases} \frac{1}{m} & x\in\mathbb{Q} \text{ with $x=\frac{n}{m}$ for $n\in \mathbb{Z}$ and $m\in \mathbb{N}$ sharing no common factors}\\
0 & x\in \mathbb{R}\setminus\mathbb{Q} \end{cases}.
\]
- (a) Show that $\mathbb{Q}$ and $\mathbb{R}\setminus\mathbb{Q}$ are dense in $\mathbb{R}$.
- (b) Show that $f$ is not continuous at any $x\in \mathbb{R}$.
- (c) Show that $g$ is not continuous at any $x\in \mathbb{Q}$.
- (d) Show that $g$ is continuous at every $x\in \mathbb{R}\setminus\mathbb{Q}$.
Homework 5, due Friday, October 9th ($\S 17, 18$) Solutions
- 1. Prove each of the following topological spaces is Hausdorff.
- (a) A set $X$ with an order relation $ < $ and the order topology.
- (b) A product $X\times Y$ with the product topology where $X$ and $Y$ are Hausdorff spaces.
- (c) A subspace $Y\subset X$ with the subspace topology where $X$ is a Hausdorff space.
- 2. Let $X$ be a topological space. Show that $X$ is Hausdorff if and only if the diagonal \[ \Delta:=\{ (x,x)\mid x\in X\} \] is a closed subset of $X\times X$ with the product topology.
- 3. Consider the collection $\mathcal{T}=\{U\subset \mathbb{R} \mid \mathbb{R}\setminus U\text{ is finite}\}\cup\{\emptyset\}$.
- (a) Show that $\mathcal{T}$ is a topology on $\mathbb{R}$. We call this the finite complement topology.
- (b) Show that the finite complement topology is $T_1$: given distinct points $x,y\in\mathbb{R}$ there exists open sets $U$ and $V$ with $x\in U\not\ni y$ and $x\not\in V\ni y$.
- (c) Show that the finite complement topology is not Hausdorff.
- (d) Find all the points that the net $(\frac1n)_{n\in\mathbb{N}}$ converges to in the finite complement topology.
- 4. Let $X$ be a set with two topologies $\mathcal{T}$ and $\mathcal{T}'$ and let $i\colon X\to X$ be the identity function: $i(x)=x$ for all $x\in X$. Equip the domain copy of $X$ with the topology $\mathcal{T}$ and the range copy of $X$ with the topology $\mathcal{T}'$.
- (a) Show that $i$ is continuous if and only if $\mathcal{T}$ is finer than $\mathcal{T}'$.
- (b) Show that $i$ is a homeomorphism if and only if $\mathcal{T}=\mathcal{T}'$.
- 5. Consider the functions $f,g\colon \mathbb{R}^2 \to \mathbb{R}$ defined by
\[
f(x,y)=x+y \qquad \text{ and }\qquad g(x,y)=x-y.
\]
- (a) Show that if $\mathbb{R}$ and $\mathbb{R}^2$ are given the standard topologies, then $f$ and $g$ are continuous.
- (b) Suppose $\mathbb{R}$ is given the lower limit topology and $\mathbb{R}^2=\mathbb{R}\times\mathbb{R}$ is given the corresponding product topology. Determine and prove the continuity or discontinuity of $f$ and $g$.
- 6*. In this exercise you will establish a homeomorphism between the following two subspaces of $\mathbb{R}^2$:
\[
X:= \mathbb{R}^2\setminus\{(0,0)\} \quad \text{ and }\qquad Y:=\{(x,y)\in\mathbb{R}^2\mid x^2 + y^2 > 1\}.
\]
Throughout, $\mathbb{R}^2$ will have the standard topology and $X$ and $Y$ will have their subspace topologies.
- (a) Define a function $\|\cdot\|\colon \mathbb{R}^2 \to [0,+\infty)$ by $\| (x,y)\| =(x^2+y^2)^{1/2}$. Show that this function is continuous when $[0,+\infty)\subset \mathbb{R}$ is given the subspace topology.
[Hint: think geometrically.] - (b) Show that $X=\{(x,y)\in\mathbb{R}^2\mid \|(x,y)\| > 0\}$ and $Y =\{(x,y)\in \mathbb{R}^2\mid \|(x,y)\| > 1\}$.
- (c) Show that $f\colon X\to \mathbb{R}^2$ defined by $f(x,y) = \frac{1}{\|(x,y)\|}(x,y)$ is continuous.
- (d) Find continuous functions $g\colon X\to Y$ and $h\colon Y\to X$ satisfying $g\circ h(x,y)=(x,y)$ and $h\circ g(x,y)=(x,y)$, and deduce that $X$ and $Y$ are homeomorphic.
- (a) Define a function $\|\cdot\|\colon \mathbb{R}^2 \to [0,+\infty)$ by $\| (x,y)\| =(x^2+y^2)^{1/2}$. Show that this function is continuous when $[0,+\infty)\subset \mathbb{R}$ is given the subspace topology.
Homework 4, due Friday, October 2nd ($\S 17$) Solutions
- 1. Let $\mathcal{C}$ be a collection of subsets of $X$. Assume that $\emptyset, X\in \mathcal{C}$ and that finite unions and arbitrary intersections of sets in $\mathcal{C}$ are in $\mathcal{C}$. Show that the collection $\mathcal{T}:=\{X\setminus C \mid C\in\mathcal{C}\}$ is a topology on $X$ and that the collection of closed sets in this topology is $\mathcal{C}$.
- 2. Let $X$ be a topological space with subset $S\subset X$. Recall that $\overline{S}$ denotes the closure of $S$ and $S^\circ$ denotes the interior of $S$. We will also denote by $S^c:=X\setminus S$ the complement of $S$.
- (a) Show that $\overline{S}=((S^c)^\circ)^c$ for all $S\subset X$.
- (b) Show that $S^\circ = (\overline{S^c})^c$ for all $S\subset X$.
- 3. Let $X$ be a topological space and let $A,B\subset X$ be subsets.
- (a) Show that $A\subset B$ implies $\overline{A}\subset \overline{B}$ and $A^\circ\subset B^\circ$.
- (b) For $A,B\subset X$, show that $\overline{A\cup B} = \overline{A}\cup\overline{B}$.
- (c) For $A,B\subset X$, show that $(A\cap B)^\circ = A^\circ \cap B^\circ$.
- (d) Let $\mathbb{R}$ have the standard topology. Find examples of subsets $A,B\subset \mathbb{R}$ such that $\overline{A\cap B}\neq \overline{A}\cap \overline{B}$ and $(A\cup B)^\circ \neq A^\circ \cup B^\circ$.
- 4. Let $X$ be a topological space. We say a subset $S\subset X$ is dense in $X$ if for every $x\in X$ and every neighborhood $U$ of $x$ one has $U\cap S\neq\emptyset$. Show the following are equivalent:
- (i) $S$ is dense in $X$.
- (ii) $(S^c)^\circ = \emptyset$.
- (iii) $\overline{S}=X$.
- 5. Let $(a_n)_{n\in \mathbb{N}}$ be a sequence of real numbers.
- (a) Show that the collection $\mathcal{F}$ of finite subsets of $\mathbb{N}$ ordered by inclusion is a directed set.
- (b) Show the following are equivalent:
- (i) The net \[ \left(\sum_{n\in F} a_n \right)_{F\in \mathcal{F}} \] converges in $\mathbb{R}$ (with the standard topology).
- (ii) For any bijection $\sigma\colon \mathbb{N}\to \mathbb{N}$, the series $\sum_{n=1}^\infty a_{\sigma(n)}$ converges.
- (iii) The series $\sum_{n=1}^\infty |a_n|$ converges.
- 6*. Let $X$ be a topological space. Define functions $C,K\colon \mathcal{P}(X)\to \mathcal{P}(X)$ by $C(A):=A^c$ and $K(A)=\overline{A}$.
- (a) Given a fixed $A\subset X$, show that successively applying $C$ and $K$ to $A$ yields at most fourteen distinct sets.
- (b) Find a subset of $\mathbb{R}$ (with the standard topology) for which fourteen distinct sets are obtained.
Homework 3, due Friday, September 25th ($\S 13, 14, 15, 16$) Solutions
- 1. Equip $\mathbb{R}$ with the standard topology. Show that a set $U\subset \mathbb{R}$ is open if and only if for all $x\in U$ there exists $\epsilon>0$ such that $(x-\epsilon, x+\epsilon)\subset U$.
- 2. Let $X$ be a space.
- (a) Let $\{\mathcal{T}_i\mid i \in I\}$ be a non-empty collection topologies on $X$ (indexed by some set $I$). Show that $\bigcap_{i\in I} \mathcal{T}_i$ is a topology on $X$.
- (b) Let $\mathcal{B}$ be a basis for a topology $\mathcal{T}$ on $X$. Show that $\mathcal{T}$ is the intersection of all topologies on $X$ that contain $\mathcal{B}$.
- (c) Let $\mathcal{S}$ be a subbasis for a topology $\mathcal{T}$ on a space $X$. Suppose $\mathcal{T}'$ is another topology on $X$ that contains $\mathcal{S}$. Show that $\mathcal{T}$ is coarser than $\mathcal{T}'$.
- (d) Let $\mathcal{S}$ and $\mathcal{T}$ be as in the previous part. Show that $\mathcal{T}$ is the intersection of all topologies on $X$ that contain $\mathcal{S}$.
- 3. Let $X$ be an ordered set (with at least two elements) equipped with the order topology. For a subspace $Y\subset X$, show that the collection $\mathcal{S}$ consisting of sets of the form $Y\cap (-\infty,a)$ or $Y\cap (a,+\infty)$ for $a\in X$ form a subbasis for the subspace topology on $Y$.
- 4. Let $X$ and $Y$ be topological spaces. A function $f\colon X\to Y$ is called an open map if for every open subset $U\subset X$ one has that its image $f(U)$ is open in $Y$.
- (a) Equip $X\times Y$ with the product topology. Show that the coordinate projections $\pi_1\colon X\times Y\to X$ and $\pi_2\colon X\times Y\to Y$ are open maps.
- (b) Let $\mathcal{B}$ be a basis for the topology on $X$ and suppose $f(B)$ is open for all $B\in \mathcal{B}$. Show that $f$ is an open map.
- (c) Show that the previous part does not hold for subbases. [Hint: consider the function $f\colon \mathbb{R}\to \mathbb{R}$ with $f(0)=1$ and $f(x)=|x|$ if $x\neq 0$ where $\mathbb{R}$ has the standard topology.]
- 5. Equip $\mathbb{R}$ with the standard topology.
- (a) Show that the subspace topology on $\{\frac1n\mid n\in\mathbb{N}\}\subset \mathbb{R}$ is the discrete topology.
- (b) Show that the subspace topology on $\{0\}\cup\{\frac1n\mid n\in\mathbb{N}\}$ is not the discrete topology.
- 6*. In this exercise, you will show that there is a countable basis that generates the standard topology on $\mathbb{R}$. For parts (a)--(c), you should only use the properties of $\mathbb{Z}$ and $\mathbb{R}$ given in $\S4$.
- (a) For $x\in \mathbb{R}$, show that there is exactly one $n\in \mathbb{Z}$ satisfying $n\leq x < n+1$.
- (b) For $x,y\in \mathbb{R}$, show that if $x-y > 1$ then there is at least one $n\in \mathbb{Z}$ satisfying $y < n < x$.
- (c) For $x,y\in \mathbb{R}$, show that if $x-y > 0$ then there exists $z\in \mathbb{Q}$ satisfying $y < z < x$.
- (d) Let $\mathcal{B}$ be the collection of open intervals $(a,b)\subset \mathbb{R}$ with $a,b\in \mathbb{Q}$. Show that $\mathcal{B}$ is countable and is a basis for a topology on $\mathbb{R}$.
- (e) Show $\mathcal{B}$ generates the standard topology on $\mathbb{R}$.
Homework 2, due Friday, September 18th ($\S 9,10,11,12$) Solutions
- 1. Let $f\colon A \to B$ be a function.
- (a) Use the axiom of choice to show that if $f$ is surjective, then there exists $g\colon B\to A$ with $f\circ g(b)=b$ for all $b\in B$.
- (b) Without using the axiom of choice show that if $f$ is injective, then there exists $h\colon B\to A$ with $h\circ f(a)=a$ for all $a\in A$.
- 2. Show that the well-ordering theorem implies the axiom of choice.
- 3. Let $S_\Omega$ be the minimal uncountable well-ordered set from $\S10$.
- (a) Show that $S_\Omega$ has no largest element.
- (b) Show that for every $x\in S_\Omega$, the subset $\{y\in S_\Omega\mid x < y\}$ is uncountable.
- (c) Consider the subset \[ X:=\{x\in S_\Omega\mid (a,x)\neq\emptyset \text{ for all } a < x\}. \] Show that $X$ is uncountable. [Hint: proceed by contradiction and use the fact that for any $y\in S_\Omega$ there exists $z\in S_\Omega$ with $(y,z)=\emptyset$.]
- 4.In this exercise you will use Zorn's lemma to prove the following fact from linear algebra: every vector space $V$ has a basis. For a subset $A\subset V$, recall: the span of $A$ is the set of all finite linear combinations of vectors in $A$; $A$ is said to be independent if the only way to write the zero vector as a linear combination of elements in $A$ is via the trivial linear combination with all zero scalar coefficients; and $A$ is said to be a basis for $V$ if it is independent and its span is all of $V$.
- (a) Suppose $A\subset V$ is independent. Show that if $v$ is not in the span of $A$, then $A\cup\{v\}$ is independent.
- (b) Show that the collection of independent subsets of $V$, ordered by inclusion, has a maximal element.
- (c) Show that $V$ has a basis.
- 5. Let $X$ be a topological space and let $A\subset X$ be a subset. Suppose that for all $x\in A$, there exists an open set $U$ satisfying $x\in U\subset A$. Show that $A$ is open.
Homework 1, due Friday, September 11th ($\S 2, 3, 6, 7$) Solutions
- 1. Let $f\colon A \to B$ be a function.
- (a) For $A_0\subset A$ and $B_0\subset B$, show that $A_0 \subset f^{-1}(f(A_0))$ and $f(f^{-1}(B_0)) \subset B_0$.
- (b) Show that $f$ is injective if and only if $A_0=f^{-1}(f(A_0))$ for all subsets $A_0\subset A$.
- (c) Show that $f$ is surjective if and only if $f(f^{-1}(B_0))=B_0$ for all subsets $B_0\subset B$.
- 2. Let $C$ be a relation on a set $A$. For a subset $A_0\subset A$, the restriction of $C$ to $A_0$ is the relation defined by the subset $D:=C\cap (A_0\times A_0)$.
- (a) For $a,b\in A$, show that $aDb$ if and only if $a,b\in A_0$ and $aCb$.
- (b) Show that if $C$ is an equivalence relation on $A$, then $D$ is an equivalence relation on $A_0$.
- (c) Show that if $C$ is an order relation on $A$, then $D$ is an order relation on $A_0$.
- (d) Show that if $C$ is a partial order relation on $A$, then $D$ is a partial order relation on $A_0$.
- 3. Let $A$ and $B$ be non-empty sets.
- (a) Prove that $A\times B$ is finite if and only if $A$ and $B$ are both finite.
- (b) Let $B^A$ denote the set of functions $f\colon A\to B$. Show that if $A$ and $B$ are finite, then so is $B^A$.
- (c) Suppose $B^A$ is finite and $B$ has at least two elements. Show that $A$ and $B$ are finite.
- 4. We say two sets $A$ and $B$ have the same cardinality if there is a bijection of $A$ with $B$. In this exercise, you will prove the Schröder–Bernstein Theorem: if there exist injections $f\colon A\to B$ and $g\colon B\to A$, then $A$ and $B$ have the same cardinality.
- (a) Suppose $C\subset A$ and that there is an injection $f\colon A\to C$. Define $A_1:=A$, $C_1:=C$, and for $n>1$ recursively define $A_n:=f(A_{n-1})$ and $C_n:=f(C_{n-1})$. Show that \[ A_1\supset C_1 \supset A_2 \supset C_2 \supset A_3\supset \cdots \] and that $f(A_n\setminus C_n)=A_{n+1}\setminus C_{n+1}$ for all $n\in \mathbb{N}$.
- (b) Using the notation from the previous part, show that $h\colon A\to C$ defined by \[ h(x):=\begin{cases} f(x) & \text{if $x\in A_n\setminus C_n$ for some $n\in \mathbb{N}$}\\ x & \text{otherwise} \end{cases} \] is a bijection. [Hint: draw a picture.]
- (c) Prove the Schröder–Bernstein Theorem.
- 5. Let $\{0,1\}^\mathbb{N}$ denote the set of functions $f\colon \mathbb{N}\to \{0,1\}$.
- (a) Show that $\{0,1\}^\mathbb{N}$ and $\mathcal{P}(\mathbb{N})$ have the same cardinality.
- (b) Let $\mathcal{C}$ be the collection of countable subsets of $\{0,1\}^\mathbb{N}$. Show that $\mathcal{C}$ and $\{0,1\}^\mathbb{N}$ have the same cardinality. [Hint: first construct an injection from $\mathcal{C}$ to $(\{0,1\}^\mathbb{N})^\mathbb{N}$ then use Exercise 4.]
