Metric And Topological Spaces

# Metric And Topological Spaces

### Jump to year

Paper 1, Section II, G

commentConsider the set of sequences of integers

$X=\left\{\left(x_{1}, x_{2}, \ldots\right) \mid x_{n} \in \mathbb{Z} \text { for all } n\right\}$

Define

$n_{\min }\left(\left(x_{n}\right),\left(y_{n}\right)\right)= \begin{cases}\infty & x_{n}=y_{n} \text { for all } n \\ \min \left\{n \mid x_{n} \neq y_{n}\right\} & \text { otherwise }\end{cases}$

for two sequences $\left(x_{n}\right),\left(y_{n}\right) \in X$. Let

$d\left(\left(x_{n}\right),\left(y_{n}\right)\right)=2^{-n_{\min }\left(\left(x_{n}\right),\left(y_{n}\right)\right)}$

where, as usual, we adopt the convention that $2^{-\infty}=0$.

(a) Prove that $d$ defines a metric on $X$.

(b) What does it mean for a metric space to be complete? Prove that $(X, d)$ is complete.

(c) Is $(X, d)$ path connected? Justify your answer.

Paper 2, Section I, G

comment(a) Let $f: X \rightarrow Y$ be a continuous surjection of topological spaces. Prove that, if $X$ is connected, then $Y$ is also connected.

(b) Let $g:[0,1] \rightarrow[0,1]$ be a continuous map. Deduce from part (a) that, for every $y$ between $g(0)$ and $g(1)$, there is $x \in[0,1]$ such that $g(x)=y$. [You may not assume the Intermediate Value Theorem, but you may use the fact that suprema exist in $\mathbb{R}$.]

Paper 3, Section I, $3 G$

commentLet $X$ be a metric space.

(a) What does it mean for $X$ to be compact? What does it mean for $X$ to be sequentially compact?

(b) Prove that if $X$ is compact then $X$ is sequentially compact.

Paper 4, Section II, G

comment(a) Define the subspace, quotient and product topologies.

(b) Let $X$ be a compact topological space and $Y$ a Hausdorff topological space. Prove that a continuous bijection $f: X \rightarrow Y$ is a homeomorphism.

(c) Let $S=[0,1] \times[0,1]$, equipped with the product topology. Let $\sim$ be the smallest equivalence relation on $S$ such that $(s, 0) \sim(s, 1)$ and $(0, t) \sim(1, t)$, for all $s, t \in[0,1]$. Let

$T=\left\{(x, y, z) \in \mathbb{R}^{3} \mid\left(\sqrt{x^{2}+y^{2}}-2\right)^{2}+z^{2}=1\right\}$

equipped with the subspace topology from $\mathbb{R}^{3}$. Prove that $S / \sim$ and $T$ are homeomorphic.

[You may assume without proof that $S$ is compact.]

Paper 1, Section II, E

commentWhat does it mean to say that a topological space is compact? Prove directly from the definition that $[0,1]$ is compact. Hence show that the unit circle $S^{1} \subset \mathbb{R}^{2}$ is compact, proving any results that you use. [You may use without proof the continuity of standard functions.]

The set $\mathbb{R}^{2}$ has a topology $\mathcal{T}$ for which the closed sets are the empty set and the finite unions of vector subspaces. Let $X$ denote the set $\mathbb{R}^{2} \backslash\{0\}$ with the subspace topology induced by $\mathcal{T}$. By considering the subspace topology on $S^{1} \subset \mathbb{R}^{2}$, or otherwise, show that $X$ is compact.

Paper 2, Section I, E

commentWhat does it mean to say that $d$ is a metric on a set $X$ ? What does it mean to say that a subset of $X$ is open with respect to the metric $d$ ? Show that the collection of subsets of $X$ that are open with respect to $d$ satisfies the axioms of a topology.

For $X=C[0,1]$, the set of continuous functions $f:[0,1] \rightarrow \mathbb{R}$, show that the metrics

$\begin{aligned} &d_{1}(f, g)=\int_{0}^{1}|f(x)-g(x)| \mathrm{d} x \\ &d_{2}(f, g)=\left[\int_{0}^{1}|f(x)-g(x)|^{2} \mathrm{~d} x\right]^{1 / 2} \end{aligned}$

give different topologies.

Paper 3, Section I, $3 E$

commentWhat does it mean to say that a topological space is connected? If $X$ is a topological space and $x \in X$, show that there is a connected subspace $K_{x}$ of $X$ so that if $S$ is any other connected subspace containing $x$ then $S \subseteq K_{x}$.

Show that the sets $K_{x}$ partition $X$.

Paper 4, Section II, E

commentLet $X=\{2,3,4,5,6,7,8, \ldots\}$ and for each $n \in X$ let

$U_{n}=\{d \in X \mid d \text { divides } n\} .$

Prove that the set of unions of the sets $U_{n}$ forms a topology on $X$.

Prove or disprove each of the following:

(i) $X$ is Hausdorff;

(ii) $X$ is compact.

If $Y$ and $Z$ are topological spaces, $Y$ is the union of closed subspaces $A$ and $B$, and $f: Y \rightarrow Z$ is a function such that both $\left.f\right|_{A}: A \rightarrow Z$ and $\left.f\right|_{B}: B \rightarrow Z$ are continuous, show that $f$ is continuous. Hence show that $X$ is path-connected.

Paper 1, Section II, E

commentConsider $\mathbb{R}$ and $\mathbb{R}^{2}$ with their usual Euclidean topologies.

(a) Show that a non-empty subset of $\mathbb{R}$ is connected if and only if it is an interval. Find a compact subset $K \subset \mathbb{R}$ for which $\mathbb{R} \backslash K$ has infinitely many connected components.

(b) Let $T$ be a countable subset of $\mathbb{R}^{2}$. Show that $\mathbb{R}^{2} \backslash T$ is path-connected. Deduce that $\mathbb{R}^{2}$ is not homeomorphic to $\mathbb{R}$.

Paper 2, Section I, E

commentLet $f:(X, d) \rightarrow(Y, e)$ be a function between metric spaces.

(a) Give the $\epsilon-\delta$ definition for $f$ to be continuous. Show that $f$ is continuous if and only if $f^{-1}(U)$ is an open subset of $X$ for each open subset $U$ of $Y$.

(b) Give an example of $f$ such that $f$ is not continuous but $f(V)$ is an open subset of $Y$ for every open subset $V$ of $X$.

Paper 3, Section I, $3 E$

commentLet $X$ and $Y$ be topological spaces.

(a) Define what is meant by the product topology on $X \times Y$. Define the projection maps $p: X \times Y \rightarrow X$ and $q: X \times Y \rightarrow Y$ and show they are continuous.

(b) Consider $\Delta=\{(x, x) \mid x \in X\}$ in $X \times X$. Show that $X$ is Hausdorff if and only if $\Delta$ is a closed subset of $X \times X$ in the product topology.

Paper 4, Section II, E

commentLet $f: X \rightarrow Y$ be a continuous map between topological spaces.

(a) Assume $X$ is compact and that $Z \subseteq X$ is a closed subset. Prove that $Z$ and $f(Z)$ are both compact.

(b) Suppose that

(i) $f^{-1}(\{y\})$ is compact for each $y \in Y$, and

(ii) if $A$ is any closed subset of $X$, then $f(A)$ is a closed subset of $Y$.

Show that if $K \subseteq Y$ is compact, then $f^{-1}(K)$ is compact.

$\left[\right.$ Hint: Given an open cover $f^{-1}(K) \subseteq \bigcup_{i \in I} U_{i}$, find a finite subcover, say $f^{-1}(\{y\}) \subseteq$ $\bigcup_{i \in I_{y}} U_{i}$, for each $y \in K$; use closedness of $X \backslash \bigcup_{i \in I_{y}} U_{i}$ and property (ii) to produce an open cover of $K$.]

Paper 1, Section II, E

commentLet $p$ be a prime number. Define what is meant by the $p$-adic metric $d_{p}$ on $\mathbb{Q}$. Show that for $a, b, c \in \mathbb{Q}$ we have

$d_{p}(a, b) \leqslant \max \left\{d_{p}(a, c), d_{p}(c, b)\right\}$

Show that the sequence $\left(a_{n}\right)$, where $a_{n}=1+p+\cdots+p^{n-1}$, converges to some element in (D.

For $a \in \mathbb{Q}$ define $|a|_{p}=d_{p}(a, 0)$. Show that if $a, b \in \mathbb{Q}$ and if $|a|_{p} \neq|b|_{p}$, then

$|a+b|_{p}=\max \left\{|a|_{p},|b|_{p}\right\} .$

Let $a \in \mathbb{Q}$ and let $B(a, \delta)$ be the open ball with centre $a$ and radius $\delta>0$, with respect to the metric $d_{p}$. Show that $B(a, \delta)$ is a closed subset of $\mathbb{Q}$ with respect to the topology induced by $d_{p}$.

Paper 2, Section I, E

commentConsider $\mathbb{R}$ and $\mathbb{Q}$ with their usual topologies.

(a) Show that compact subsets of a Hausdorff topological space are closed. Show that compact subsets of $\mathbb{R}$ are closed and bounded.

(b) Show that there exists a complete metric space $(X, d)$ admitting a surjective continuous map $f: X \rightarrow \mathbb{Q}$.

Paper 3, Section I, $3 E$

commentLet $X$ be a topological space and $A \subseteq X$ be a subset. A limit point of $A$ is a point $x \in X$ such that any open neighbourhood $U$ of $x$ intersects $A$. Show that $A$ is closed if and only if it contains all its limit points. Explain what is meant by the interior Int $(A)$ and the closure $\bar{A}$ of $A$. Show that if $A$ is connected, then $\bar{A}$ is connected.

Paper 4, Section II, E

comment(a) Let $X$ be a topological space. Define what is meant by a quotient of $X$ and describe the quotient topology on the quotient space. Give an example in which $X$ is Hausdorff but the quotient space is not Hausdorff.

(b) Let $T^{2}$ be the 2-dimensional torus considered as the quotient $\mathbb{R}^{2} / \mathbb{Z}^{2}$, and let $\pi: \mathbb{R}^{2} \rightarrow T^{2}$ be the quotient map.

(i) Let $B(u, r)$ be the open ball in $\mathbb{R}^{2}$ with centre $u$ and radius $r<1 / 2$. Show that $U=\pi(B(u, r))$ is an open subset of $T^{2}$ and show that $\pi^{-1}(U)$ has infinitely many connected components. Show each connected component is homeomorphic to $B(u, r)$.

(ii) Let $\alpha \in \mathbb{R}$ be an irrational number and let $L \subset \mathbb{R}^{2}$ be the line given by the equation $y=\alpha x$. Show that $\pi(L)$ is dense in $T^{2}$ but $\pi(L) \neq T^{2}$.

Paper 1, Section II, E

commentGive the definition of a metric on a set $X$ and explain how this defines a topology on $X$.

Suppose $(X, d)$ is a metric space and $U$ is an open set in $X$. Let $x, y \in X$ and $\epsilon>0$ such that the open ball $B_{\epsilon}(y) \subseteq U$ and $x \in B_{\epsilon / 2}(y)$. Prove that $y \in B_{\epsilon / 2}(x) \subseteq U$.

Explain what it means (i) for a set $S$ to be dense in $X$, (ii) to say $\mathcal{B}$ is a base for a topology $\mathcal{T}$.

Prove that any metric space which contains a countable dense set has a countable basis.

Paper 2, Section I, $4 \mathbf{E}$

commentLet $X$ and $Y$ be topological spaces and $f: X \rightarrow Y$ a continuous map. Suppose $H$ is a subset of $X$ such that $f(\bar{H})$ is closed (where $\bar{H}$ denotes the closure of $H$ ). Prove that $f(\bar{H})=\overline{f(H)} .$

Give an example where $f, X, Y$ and $H$ are as above but $f(\bar{H})$ is not closed.

Paper 3, Section I, $3 \mathrm{E}$

commentDefine what it means for a topological space $X$ to be (i) connected (ii) path-connected.

Prove that any path-connected space $X$ is connected. [You may assume the interval $[0,1]$ is connected. $]$

Give a counterexample (without justification) to the converse statement.

Paper 4, Section II, E

commentExplain what it means for a metric space $(M, d)$ to be (i) compact, (ii) sequentially compact. Prove that a compact metric space is sequentially compact, stating clearly any results that you use.

Let $(M, d)$ be a compact metric space and suppose $f: M \rightarrow M$ satisfies $d(f(x), f(y))=d(x, y)$ for all $x, y \in M$. Prove that $f$ is surjective, stating clearly any results that you use. [Hint: Consider the sequence $\left(f^{n}(x)\right)$ for $x \in M$.]

Give an example to show that the result does not hold if $M$ is not compact.

Paper 1, Section II, E

commentDefine what it means for a topological space to be compact. Define what it means for a topological space to be Hausdorff.

Prove that a compact subspace of a Hausdorff space is closed. Hence prove that if $C_{1}$ and $C_{2}$ are compact subspaces of a Hausdorff space $X$ then $C_{1} \cap C_{2}$ is compact.

A subset $U$ of $\mathbb{R}$ is open in the cocountable topology if $U$ is empty or its complement in $\mathbb{R}$ is countable. Is $\mathbb{R}$ Hausdorff in the cocountable topology? Which subsets of $\mathbb{R}$ are compact in the cocountable topology?

Paper 2, Section $I$, E

commentLet $X$ and $Y$ be topological spaces. What does it mean to say that a function $f: X \rightarrow Y$ is continuous?

Are the following statements true or false? Give proofs or counterexamples.

(i) Every continuous function $f: X \rightarrow Y$ is an open map, i.e. if $U$ is open in $X$ then $f(U)$ is open in $Y$.

(ii) If $f: X \rightarrow Y$ is continuous and bijective then $f$ is a homeomorphism.

(iii) If $f: X \rightarrow Y$ is continuous, open and bijective then $f$ is a homeomorphism.

Paper 3, Section I, E

commentSuppose $(X, d)$ is a metric space. Do the following necessarily define a metric on $X$ ? Give proofs or counterexamples.

(i) $d_{1}(x, y)=k d(x, y)$ for some constant $k>0$, for all $x, y \in X$.

(ii) $d_{2}(x, y)=\min \{1, d(x, y)\}$ for all $x, y \in X$.

(iii) $d_{3}(x, y)=(d(x, y))^{2}$ for all $x, y \in X$.

Paper 4, Section II, E

commentExplain what it means for a metric space to be complete.

Let $X$ be a metric space. We say the subsets $A_{i}$ of $X$, with $i \in \mathbb{N}$, form a descending sequence in $X$ if $A_{1} \supset A_{2} \supset A_{3} \supset \cdots$.

Prove that the metric space $X$ is complete if and only if any descending sequence $A_{1} \supset A_{2} \supset \cdots$ of non-empty closed subsets of $X$, such that the diameters of the subsets $A_{i}$ converge to zero, has an intersection $\bigcap_{i=1}^{\infty} A_{i}$ that is non-empty.

[Recall that the diameter $\operatorname{diam}(S)$ of a set $S$ is the supremum of the set $\{d(x, y)$ : $x, y \in S\} .]$

Give examples of

(i) a metric space $X$, and a descending sequence $A_{1} \supset A_{2} \supset \cdots$ of non-empty closed subsets of $X$, with $\operatorname{diam}\left(A_{i}\right)$ converging to 0 but $\bigcap_{i=1}^{\infty} A_{i}=\emptyset$.

(ii) a descending sequence $A_{1} \supset A_{2} \supset \cdots$ of non-empty sets in $\mathbb{R}$ with $\operatorname{diam}\left(A_{i}\right)$ converging to 0 but $\bigcap_{i=1}^{\infty} A_{i}=\emptyset$.

(iii) a descending sequence $A_{1} \supset A_{2} \supset \cdots$ of non-empty closed sets in $\mathbb{R}$ with $\bigcap_{i=1}^{\infty} A_{i}=\emptyset$.

Paper 1, Section II, G

commentConsider the sphere $S^{2}=\left\{(x, y, z) \in \mathbb{R}^{3} \mid x^{2}+y^{2}+z^{2}=1\right\}$, a subset of $\mathbb{R}^{3}$, as a subspace of $\mathbb{R}^{3}$ with the Euclidean metric.

(i) Show that $S^{2}$ is compact and Hausdorff as a topological space.

(ii) Let $X=S^{2} / \sim$ be the quotient set with respect to the equivalence relation identifying the antipodes, i.e.

$(x, y, z) \sim\left(x^{\prime}, y^{\prime}, z^{\prime}\right) \Longleftrightarrow\left(x^{\prime}, y^{\prime}, z^{\prime}\right)=(x, y, z) \text { or }(-x,-y,-z)$

Show that $X$ is compact and Hausdorff with respect to the quotient topology.

Paper 2, Section I, G

commentLet $X$ be a topological space. Prove or disprove the following statements.

(i) If $X$ is discrete, then $X$ is compact if and only if it is a finite set.

(ii) If $Y$ is a subspace of $X$ and $X, Y$ are both compact, then $Y$ is closed in $X$.

Paper 3, Section I, G

commentLet $X$ be a metric space with the metric $d: X \times X \rightarrow \mathbb{R}$.

(i) Show that if $X$ is compact as a topological space, then $X$ is complete.

(ii) Show that the completeness of $X$ is not a topological property, i.e. give an example of two metrics $d, d^{\prime}$ on a set $X$, such that the associated topologies are the same, but $(X, d)$ is complete and $\left(X, d^{\prime}\right)$ is not.

Paper 4, Section II, G

commentLet $X$ be a topological space. A connected component of $X$ means an equivalence class with respect to the equivalence relation on $X$ defined as:

$x \sim y \Longleftrightarrow x, y \text { belong to some connected subspace of } X .$

(i) Show that every connected component is a connected and closed subset of $X$.

(ii) If $X, Y$ are topological spaces and $X \times Y$ is the product space, show that every connected component of $X \times Y$ is a direct product of connected components of $X$ and $Y$.

Paper 1, Section II, F

commentA topological space $X$ is said to be normal if each point of $X$ is a closed subset of $X$ and for each pair of closed sets $C_{1}, C_{2} \subset X$ with $C_{1} \cap C_{2}=\emptyset$ there are open sets $U_{1}, U_{2} \subset X$ so that $C_{i} \subset U_{i}$ and $U_{1} \cap U_{2}=\emptyset$. In this case we say that the $U_{i}$ separate the $C_{i}$.

Show that a compact Hausdorff space is normal. [Hint: first consider the case where $C_{2}$ is a point.]

For $C \subset X$ we define an equivalence relation $\sim_{C}$ on $X$ by $x \sim_{C} y$ for all $x, y \in C$, $x \sim_{C} x$ for $x \notin C$. If $C, C_{1}$ and $C_{2}$ are pairwise disjoint closed subsets of a normal space $X$, show that $C_{1}$ and $C_{2}$ may be separated by open subsets $U_{1}$ and $U_{2}$ such that $U_{i} \cap C=\emptyset$. Deduce that the quotient space $X / \sim_{C}$ is also normal.

Paper 2, Section I,

commentFor each case below, determine whether the given metrics $d_{1}$ and $d_{2}$ induce the same topology on $X$. Justify your answers.

$\begin{gathered} \text { (i) } X=\mathbb{R}^{2}, d_{1}\left(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\right)=\sup \left\{\left|x_{1}-x_{2}\right|,\left|y_{1}-y_{2}\right|\right\} \\ d_{2}\left(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\right)=\left|x_{1}-x_{2}\right|+\left|y_{1}-y_{2}\right| . \\ \text { (ii) } X=C[0,1], d_{1}(f, g)=\sup _{t \in[0,1]}|f(t)-g(t)| \\ d_{2}(f, g)=\int_{0}^{1}|f(t)-g(t)| d t . \end{gathered}$

Paper 3, Section I, $3 F$

commentDefine the notion of a connected component of a space $X$.

If $A_{\alpha} \subset X$ are connected subsets of $X$ such that $\bigcap_{\alpha} A_{\alpha} \neq \emptyset$, show that $\bigcup_{\alpha} A_{\alpha}$ is connected.

Prove that any point $x \in X$ is contained in a unique connected component.

Let $X \subset \mathbb{R}$ consist of the points $0,1, \frac{1}{2}, \frac{1}{3}, \ldots, \frac{1}{n}, \ldots$. What are the connected components of $X$ ?

Paper 4, Section II, F

commentSuppose $A_{1}$ and $A_{2}$ are topological spaces. Define the product topology on $A_{1} \times A_{2}$. Let $\pi_{i}: A_{1} \times A_{2} \rightarrow A_{i}$ be the projection. Show that a map $F: X \rightarrow A_{1} \times A_{2}$ is continuous if and only if $\pi_{1} \circ F$ and $\pi_{2} \circ F$ are continuous.

Prove that if $A_{1}$ and $A_{2}$ are connected, then $A_{1} \times A_{2}$ is connected.

Let $X$ be the topological space whose underlying set is $\mathbb{R}$, and whose open sets are of the form $(a, \infty)$ for $a \in \mathbb{R}$, along with the empty set and the whole space. Describe the open sets in $X \times X$. Are the maps $f, g: X \times X \rightarrow X$ defined by $f(x, y)=x+y$ and $g(x, y)=x y$ continuous? Justify your answers.

Paper 1, Section II, G

commentLet $X$ be a metric space with the distance function $d: X \times X \rightarrow \mathbb{R}$. For a subset $Y$ of $X$, its diameter is defined as $\delta(Y):=\sup \left\{d\left(y, y^{\prime}\right) \mid y, y^{\prime} \in Y\right\}$.

Show that, if $X$ is compact and $\left\{U_{\lambda}\right\}_{\lambda \in \Lambda}$ is an open covering of $X$, then there exists an $\epsilon>0$ such that every subset $Y \subset X$ with $\delta(Y)<\epsilon$ is contained in some $U_{\lambda}$.

Paper 2, Section I, 4G

comment(i) Let $t>0$. For $\mathbf{x}=(x, y), \mathbf{x}^{\prime}=\left(x^{\prime}, y^{\prime}\right) \in \mathbb{R}^{2}$, let

$\begin{gathered} d\left(\mathbf{x}, \mathbf{x}^{\prime}\right)=\left|x^{\prime}-x\right|+t\left|y^{\prime}-y\right|, \\ \delta\left(\mathbf{x}, \mathbf{x}^{\prime}\right)=\sqrt{\left(x^{\prime}-x\right)^{2}+\left(y^{\prime}-y\right)^{2}} \end{gathered}$

( $\delta$ is the usual Euclidean metric on $\mathbb{R}^{2}$.) Show that $d$ is a metric on $\mathbb{R}^{2}$ and that the two metrics $d, \delta$ give rise to the same topology on $\mathbb{R}^{2}$.

(ii) Give an example of a topology on $\mathbb{R}^{2}$, different from the one in (i), whose induced topology (subspace topology) on the $x$-axis is the usual topology (the one defined by the metric $\left.d\left(x, x^{\prime}\right)=\left|x^{\prime}-x\right|\right)$. Justify your answer.

Paper 3, Section I, 3G

commentLet $X, Y$ be topological spaces, and suppose $Y$ is Hausdorff.

(i) Let $f, g: X \rightarrow Y$ be two continuous maps. Show that the set

$E(f, g):=\{x \in X \mid f(x)=g(x)\} \subset X$

is a closed subset of $X$.

(ii) Let $W$ be a dense subset of $X$. Show that a continuous map $f: X \rightarrow Y$ is determined by its restriction $\left.f\right|_{W}$ to $W$.

Paper 4, Section II, 13G

commentLet $X, Y$ be topological spaces and $X \times Y$ their product set. Let $p_{Y}: X \times Y \rightarrow Y$ be the projection map.

(i) Define the product topology on $X \times Y$. Prove that if a subset $Z \subset X \times Y$ is open then $p_{Y}(Z)$ is open in $Y$.

(ii) Give an example of $X, Y$ and a closed set $Z \subset X \times Y$ such that $p_{Y}(Z)$ is not closed.

(iii) When $X$ is compact, show that if a subset $Z \subset X \times Y$ is closed then $p_{Y}(Z)$ is closed

Paper 1, Section II, H

commentLet $f: X \rightarrow Y$ and $g: Y \rightarrow X$ be continuous maps of topological spaces with $f \circ g=\mathrm{id}_{Y}$.

(1) Suppose that (i) $Y$ is path-connected, and (ii) for every $y \in Y$, its inverse image $f^{-1}(y)$ is path-connected. Prove that $X$ is path-connected.

(2) Prove the same statement when "path-connected" is everywhere replaced by "connected".

Paper 2, Section I, H

commentOn the set $\mathbb{Q}$ of rational numbers, the 3 -adic metric $d_{3}$ is defined as follows: for $x, y \in \mathbb{Q}$, define $d_{3}(x, x)=0$ and $d_{3}(x, y)=3^{-n}$, where $n$ is the integer satisfying $x-y=3^{n} u$ where $u$ is a rational number whose denominator and numerator are both prime to 3 .

(1) Show that this is indeed a metric on $\mathbb{Q}$.

(2) Show that in $\left(\mathbb{Q}, d_{3}\right)$, we have $3^{n} \rightarrow 0$ as $n \rightarrow \infty$ while $3^{-n} \nrightarrow \infty$ as $n \rightarrow \infty$. Let $d$ be the usual metric $d(x, y)=|x-y|$ on $\mathbb{Q}$. Show that neither the identity map $(\mathbb{Q}, d) \rightarrow\left(\mathbb{Q}, d_{3}\right)$ nor its inverse is continuous.

Paper 3, Section I, H

commentLet $X$ be a topological space and $Y$ be a set. Let $p: X \rightarrow Y$ be a surjection. The quotient topology on $Y$ is defined as follows: a subset $V \subset Y$ is open if and only if $p^{-1}(V)$ is open in $X$.

(1) Show that this does indeed define a topology on $Y$, and show that $p$ is continuous when we endow $Y$ with this topology.

(2) Let $Z$ be another topological space and $f: Y \rightarrow Z$ be a map. Show that $f$ is continuous if and only if $f \circ p: X \rightarrow Z$ is continuous.

Paper 4, Section II, H

comment(1) Prove that if $X$ is a compact topological space, then a closed subset $Y$ of $X$ endowed with the subspace topology is compact.

(2) Consider the following equivalence relation on $\mathbb{R}^{2}$ :

$\left(x_{1}, y_{1}\right) \sim\left(x_{2}, y_{2}\right) \Longleftrightarrow\left(x_{1}-x_{2}, y_{1}-y_{2}\right) \in \mathbb{Z}^{2}$

Let $X=\mathbb{R}^{2} / \sim$ be the quotient space endowed with the quotient topology, and let $p: \mathbb{R}^{2} \rightarrow X$ be the canonical surjection mapping each element to its equivalence class. Let $Z=\left\{(x, y) \in \mathbb{R}^{2} \mid y=\sqrt{2} x\right\} .$

(i) Show that $X$ is compact.

(ii) Assuming that $p(Z)$ is dense in $X$, show that $\left.p\right|_{Z}: Z \rightarrow p(Z)$ is bijective but not homeomorphic.

Paper 1, Section II, F

commentGiven a function $f: X \rightarrow Y$ between metric spaces, we write $\Gamma_{f}$ for the subset $\{(x, f(x)) \mid x \in X\}$ of $X \times Y .$

(i) If $f$ is continuous, show that $\Gamma_{f}$ is closed in $X \times Y$.

(ii) If $Y$ is compact and $\Gamma_{f}$ is closed in $X \times Y$, show that $f$ is continuous.

(iii) Give an example of a function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $\Gamma_{f}$ is closed but $f$ is not continuous.

Paper 2, Section I, $4 F$

commentExplain what is meant by a Hausdorff (topological) space, and prove that every compact subset of a Hausdorff space is closed.

Let $X$ be an uncountable set, and consider the topology $\mathcal{T}$ on $X$ defined by

$U \in \mathcal{T} \Leftrightarrow \text { either } U=\emptyset \text { or } X \backslash U \text { is countable. }$

Is $(X, \mathcal{T})$ Hausdorff? Is every compact subset of $X$ closed? Justify your answers.

Paper 3, Section I, F

commentAre the following statements true or false? Give brief justifications for your answers.

(i) If $X$ is a connected open subset of $\mathbb{R}^{n}$ for some $n$, then $X$ is path-connected.

(ii) A cartesian product of two connected spaces is connected.

(iii) If $X$ is a Hausdorff space and the only connected subsets of $X$ are singletons $\{x\}$, then $X$ is discrete.

Paper 4, Section II, F

commentA nonempty subset $A$ of a topological space $X$ is called irreducible if, whenever we have open sets $U$ and $V$ such that $U \cap A$ and $V \cap A$ are nonempty, then we also have $U \cap V \cap A \neq \emptyset$. Show that the closure of an irreducible set is irreducible, and deduce that the closure of any singleton set $\{x\}$ is irreducible.

$X$ is said to be a sober topological space if, for any irreducible closed $A \subseteq X$, there is a unique $x \in X$ such that $A=\overline{\{x\}}$. Show that any Hausdorff space is sober, but that an infinite set with the cofinite topology is not sober.

Given an arbitrary topological space $(X, \mathcal{T})$, let $\widehat{X}$denote the set of all irreducible closed subsets of $X$, and for each $U \in \mathcal{T}$ let

$\widehat{U}=\{A \in \widehat{X} \mid U \cap A \neq \emptyset\}$

Show that the sets $\{\widehat{U} \mid U \in \mathcal{T}\}$ form a topology $\widehat{\mathcal{T}}$on $\widehat{X}$, and that the mapping $U \mapsto \widehat{U}$is a bijection from $\mathcal{T}$ to $\widehat{\mathcal{T}}$. Deduce that $(\widehat{X}, \widehat{\mathcal{T}}$) is sober. [Hint: consider the complement of an irreducible closed subset of $\widehat{X}$.]

$3 . \mathrm{I} . 4 \mathrm{~F} \quad$

commentExplain what it means for a topological space to be connected.

Are the following subspaces of the unit square $[0,1] \times[0,1]$ connected? Justify your answers.

(a) $\{(x, y): x \neq 0, y \neq 0$, and $x / y \in \mathbb{Q}\}$.

(b) $\{(x, y):(x=0)$ or $(x \neq 0$ and $y \in \mathbb{Q})\}$.

1.II.12F

commentWrite down the definition of a topology on a set $X$.

For each of the following families $\mathcal{T}$ of subsets of $\mathbb{Z}$, determine whether $\mathcal{T}$ is a topology on $\mathbb{Z}$. In the cases where the answer is 'yes', determine also whether $(\mathbb{Z}, \mathcal{T})$ is a Hausdorff space and whether it is compact.

(a) $\mathcal{T}=\{U \subseteq \mathbb{Z}$ : either $U$ is finite or $0 \in U\}$.

(b) $\mathcal{T}=\{U \subseteq \mathbb{Z}$ : either $\mathbb{Z} \backslash U$ is finite or $0 \notin U\}$.

(c) $\mathcal{T}=\{U \subseteq \mathbb{Z}$ : there exists $k>0$ such that, for all $n, n \in U \Leftrightarrow n+k \in U\}$.

(d) $\mathcal{T}=\{U \subseteq \mathbb{Z}$ : for all $n \in U$, there exists $k>0$ such that $\{n+k m: m \in \mathbb{Z}\} \subseteq U\}$.

2.I.4F

commentStating carefully any results on compactness which you use, show that if $X$ is a compact space, $Y$ is a Hausdorff space and $f: X \rightarrow Y$ is bijective and continuous, then $f$ is a homeomorphism.

Hence or otherwise show that the unit circle $S=\left\{(x, y) \in \mathbb{R}^{2}: x^{2}+y^{2}=1\right\}$ is homeomorphic to the quotient space $[0,1] / \sim$, where $\sim$ is the equivalence relation defined by

$x \sim y \Leftrightarrow \text { either } x=y \text { or }\{x, y\}=\{0,1\} .$

4.II.14F

commentExplain what is meant by a base for a topology. Illustrate your definition by describing bases for the topology induced by a metric on a set, and for the product topology on the cartesian product of two topological spaces.

A topological space $(X, \mathcal{T})$ is said to be separable if there is a countable subset $C \subseteq X$ which is dense, i.e. such that $C \cap U \neq \emptyset$ for every nonempty $U \in \mathcal{T}$. Show that a product of two separable spaces is separable. Show also that a metric space is separable if and only if its topology has a countable base, and deduce that every subspace of a separable metric space is separable.

Now let $X=\mathbb{R}$ with the topology $\mathcal{T}$ having as a base the set of all half-open intervals

$[a, b)=\{x \in \mathbb{R}: a \leqslant x<b\}$

with $a<b$. Show that $X$ is separable, but that the subspace $Y=\{(x,-x): x \in \mathbb{R}\}$ of $X \times X$ is not separable.

[You may assume standard results on countability.]

1.II.12A

commentLet $X$ and $Y$ be topological spaces. Define the product topology on $X \times Y$ and show that if $X$ and $Y$ are Hausdorff then so is $X \times Y$.

Show that the following statements are equivalent.

(i) $X$ is a Hausdorff space.

(ii) The diagonal $\Delta=\{(x, x): x \in X\}$ is a closed subset of $X \times X$, in the product topology.

(iii) For any topological space $Y$ and any continuous maps $f, g: Y \rightarrow X$, the set $\{y \in Y: f(y)=g(y)\}$ is closed in $Y$.

2.I.4A

commentAre the following statements true or false? Give a proof or a counterexample as appropriate.

(i) If $f: X \rightarrow Y$ is a continuous map of topological spaces and $S \subseteq X$ is compact then $f(S)$ is compact.

(ii) If $f: X \rightarrow Y$ is a continuous map of topological spaces and $K \subseteq Y$ is compact then $\left.f^{-1}(K)=\{x \in X: f(x) \in K\}\right\}$ is compact.

(iii) If a metric space $M$ is complete and a metric space $T$ is homeomorphic to $M$ then $T$ is complete.

3.I.4A

comment(a) Let $X$ be a connected topological space such that each point $x$ of $X$ has a neighbourhood homeomorphic to $\mathbb{R}^{n}$. Prove that $X$ is path-connected.

(b) Let $\tau$ denote the topology on $\mathbb{N}=\{1,2, \ldots\}$, such that the open sets are $\mathbb{N}$, the empty set, and all the sets $\{1,2, \ldots, n\}$, for $n \in \mathbb{N}$. Prove that any continuous map from the topological space $(\mathbb{N}, \tau)$ to the Euclidean $\mathbb{R}$ is constant.

4.II.14A

comment(a) For a subset $A$ of a topological space $X$, define the closure cl $(A)$ of $A$. Let $f: X \rightarrow Y$ be a map to a topological space $Y$. Prove that $f$ is continuous if and only if $f(c l(A)) \subseteq c l(f(A))$, for each $A \subseteq X$.

(b) Let $M$ be a metric space. A subset $S$ of $M$ is called dense in $M$ if the closure of $S$ is equal to $M$.

Prove that if a metric space $M$ is compact then it has a countable subset which is dense in $M$.

1.II.12F

comment(i) Define the product topology on $X \times Y$ for topological spaces $X$ and $Y$, proving that your definition does define a topology.

(ii) Let $X$ be the logarithmic spiral defined in polar coordinates by $r=e^{\theta}$, where $-\infty<\theta<\infty$. Show that $X$ (with the subspace topology from $\mathbf{R}^{2}$ ) is homeomorphic to the real line.

2.I.4F

commentWhich of the following subspaces of Euclidean space are connected? Justify your answers (i) $\left\{(x, y, z) \in \mathbf{R}^{3}: z^{2}-x^{2}-y^{2}=1\right\}$; (ii) $\left\{(x, y) \in \mathbf{R}^{2}: x^{2}=y^{2}\right\}$; (iii) $\left\{(x, y, z) \in \mathbf{R}^{3}: z=x^{2}+y^{2}\right\}$.

3.I.4F

commentWhich of the following are topological spaces? Justify your answers.

(i) The set $X=\mathbf{Z}$ of the integers, with a subset $A$ of $X$ called "open" when $A$ is either finite or the whole set $X$;

(ii) The set $X=\mathbf{Z}$ of the integers, with a subset $A$ of $X$ called "open" when, for each element $x \in A$ and every even integer $n, x+n$ is also in $A$

4.II.14F

comment(a) Show that every compact subset of a Hausdorff topological space is closed.

(b) Let $X$ be a compact metric space. For $F$ a closed subset of $X$ and $p$ any point of $X$, show that there is a point $q$ in $F$ with

$d(p, q)=\inf _{q^{\prime} \in F} d\left(p, q^{\prime}\right)$

Suppose that for every $x$ and $y$ in $X$ there is a point $m$ in $X$ with $d(x, m)=(1 / 2) d(x, y)$ and $d(y, m)=(1 / 2) d(x, y)$. Show that $X$ is connected.

1.II.12A

commentSuppose that $\left(X, d_{X}\right)$ and $\left(Y, d_{Y}\right)$ are metric spaces. Show that the definition

$d\left(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\right)=d_{X}\left(x_{1}, x_{2}\right)+d_{Y}\left(y_{1}, y_{2}\right)$

defines a metric on the product $X \times Y$, under which the projection map $\pi: X \times Y \rightarrow Y$ is continuous.

If $\left(X, d_{X}\right)$ is compact, show that every sequence in $X$ has a subsequence converging to a point of $X$. Deduce that the projection map $\pi$ then has the property that, for any closed subset $F \subset X \times Y$, the image $\pi(F)$ is closed in $Y$. Give an example to show that this fails if $\left(X, d_{X}\right)$ is not assumed compact.

2.I.4A

commentLet $X$ be a topological space. Suppose that $U_{1}, U_{2}, \ldots$ are connected subsets of $X$ with $U_{j} \cap U_{1}$ non-empty for all $j>0$. Prove that

$W=\bigcup_{j>0} U_{j}$

is connected. If each $U_{j}$ is path-connected, prove that $W$ is path-connected.

3.I.4A

commentShow that a topology $\tau_{1}$ is determined on the real line $\mathbf{R}$ by specifying that a nonempty subset is open if and only if it is a union of half-open intervals $\{a \leq x<b\}$, where $a<b$ are real numbers. Determine whether $\left(\mathbf{R}, \tau_{1}\right)$ is Hausdorff.

Let $\tau_{2}$ denote the cofinite topology on $\mathbf{R}$ (that is, a non-empty subset is open if and only if its complement is finite). Prove that the identity map induces a continuous $\operatorname{map}\left(\mathbf{R}, \tau_{1}\right) \rightarrow\left(\mathbf{R}, \tau_{2}\right)$.

4.II.14A

commentLet $(M, d)$ be a metric space, and $F$ a non-empty closed subset of $M$. For $x \in M$, set

$d(x, F)=\inf _{z \in F} d(x, z)$

Prove that $d(x, F)$ is a continuous function of $x$, and that it is strictly positive for $x \notin F$.

A topological space is called normal if for any pair of disjoint closed subsets $F_{1}, F_{2}$, there exist disjoint open subsets $U_{1} \supset F_{1}, U_{2} \supset F_{2}$. By considering the function

$d\left(x, F_{1}\right)-d\left(x, F_{2}\right)$

or otherwise, deduce that any metric space is normal.

Suppose now that $X$ is a normal topological space, and that $F_{1}, F_{2}$ are disjoint closed subsets in $X$. Prove that there exist open subsets $W_{1} \supset F_{1}, W_{2} \supset F_{2}$, whose closures are disjoint. In the case when $X=\mathbf{R}^{2}$ with the standard metric topology, $F_{1}=\{(x,-1 / x): x<0\}$ and $F_{2}=\{(x, 1 / x): x>0\}$, find explicit open subsets $W_{1}, W_{2}$ with the above property.