(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 $\mathbb{R}$ .]

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Paper 3, Section I, $3 G$

Metric and Topological Spaces | Part IB, 2019

Let $X$ 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.

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Paper 4, Section II, G

Metric and Topological Spaces | Part IB, 2019

(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.]

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Paper 1, Section II, E

Metric and Topological Spaces | Part IB, 2018

What 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.

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Paper 2, Section I, E

Metric and Topological Spaces | Part IB, 2018

What 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.

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Paper 3, Section I, $3 E$

Metric and Topological Spaces | Part IB, 2018

What 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 $X$ .

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Paper 4, Section II, E

Metric and Topological Spaces | Part IB, 2018

Let $X=\{2,3,4,5,6,7,8, \ldots\}$ 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.

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Paper 1, Section II, E

Metric and Topological Spaces | Part IB, 2017

Consider $\mathbb{R}$ 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 $\mathbb{R}$ .

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Paper 2, Section I, E

Metric and Topological Spaces | Part IB, 2017

Let $f:(X, d) \rightarrow(Y, e)$ 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 $X$ .

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Paper 3, Section I, $3 E$

Metric and Topological Spaces | Part IB, 2017

Let $X$ 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.

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Paper 4, Section II, E

Metric and Topological Spaces | Part IB, 2017

Let $f: X \rightarrow Y$ 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 $K$ .]

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Paper 1, Section II, E

Metric and Topological Spaces | Part IB, 2016

Let $p$ 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 $d_{p}$ .

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Paper 2, Section I, E

Metric and Topological Spaces | Part IB, 2016

Consider $\mathbb{R}$ 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 $f: X \rightarrow \mathbb{Q}$ .

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Paper 3, Section I, $3 E$

Metric and Topological Spaces | Part IB, 2016

Let $X$ 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.

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Paper 4, Section II, E

Metric and Topological Spaces | Part IB, 2016

(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 $\pi(L) \neq T^{2}$ .

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Paper 1, Section II, E

Metric and Topological Spaces | Part IB, 2015

Give 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.

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

Metric and Topological Spaces | Part IB, 2015

Let $X$ 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.

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

Metric and Topological Spaces | Part IB, 2015

Define 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.

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Paper 4, Section II, E

Metric and Topological Spaces | Part IB, 2015

Explain 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.

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Paper 1, Section II, E

Metric and Topological Spaces | Part IB, 2014

Define 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?

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Paper 2, Section $I$ , E

Metric and Topological Spaces | Part IB, 2014

Let $X$ 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.

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Paper 3, Section I, E

Metric and Topological Spaces | Part IB, 2014

Suppose $(X, d)$ 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 $x, y \in X$ .

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Paper 4, Section II, E

Metric and Topological Spaces | Part IB, 2014

Explain 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 $\bigcap_{i=1}^{\infty} A_{i}=\emptyset$ .

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Paper 1, Section II, G

Metric and Topological Spaces | Part IB, 2013

Consider 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.

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Paper 2, Section I, G

Metric and Topological Spaces | Part IB, 2013

Let $X$ 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 $X$ .

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Paper 3, Section I, G

Metric and Topological Spaces | Part IB, 2013

Let $X$ 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.

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Paper 4, Section II, G

Metric and Topological Spaces | Part IB, 2013

Let $X$ 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 $Y$ .

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Paper 1, Section II, F

Metric and Topological Spaces | Part IB, 2012

A 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.

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Paper 2, Section I,

Metric and Topological Spaces | Part IB, 2012

For 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}

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Paper 3, Section I, $3 F$

Metric and Topological Spaces | Part IB, 2012

Define 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 $X$ ?

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Paper 4, Section II, F

Metric and Topological Spaces | Part IB, 2012

Suppose $A_{1}$ 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.

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Paper 1, Section II, G

Metric and Topological Spaces | Part IB, 2011

Let $X$ 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 $U_{\lambda}$ .

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Paper 2, Section I, 4G

Metric and Topological Spaces | Part IB, 2011

(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.

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Paper 3, Section I, 3G

Metric and Topological Spaces | Part IB, 2011

Let $X, Y$ 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 $W$ .

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Paper 4, Section II, 13G

Metric and Topological Spaces | Part IB, 2011

Let $X, Y$ 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

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Paper 1, Section II, H

Metric and Topological Spaces | Part IB, 2010

Let $f: X \rightarrow Y$ 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".

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Paper 2, Section I, H

Metric and Topological Spaces | Part IB, 2010

On 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.

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Paper 3, Section I, H

Metric and Topological Spaces | Part IB, 2010

Let $X$ 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.

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Paper 4, Section II, H

Metric and Topological Spaces | Part IB, 2010

(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.

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Paper 1, Section II, F

Metric and Topological Spaces | Part IB, 2009

Given 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.

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Paper 2, Section I, $4 F$

Metric and Topological Spaces | Part IB, 2009

Explain 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.

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Paper 3, Section I, F

Metric and Topological Spaces | Part IB, 2009

Are 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.

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Paper 4, Section II, F

Metric and Topological Spaces | Part IB, 2009

A 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 $\widehat{X}$ .]

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

Metric and Topological Spaces | Part IB, 2008

Explain 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 $y \in \mathbb{Q})\}$ .

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1.II.12F

Metric and Topological Spaces | Part IB, 2008

Write 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 $\{n+k m: m \in \mathbb{Z}\} \subseteq U\}$ .

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2.I.4F

Metric and Topological Spaces | Part IB, 2008

Stating 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\} .

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4.II.14F

Metric and Topological Spaces | Part IB, 2008

Explain 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.]

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1.II.12A

Metric and Topological Spaces | Part IB, 2007

Let $X$ 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 $Y$ .

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2.I.4A

Metric and Topological Spaces | Part IB, 2007

Are 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.

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3.I.4A

Metric and Topological Spaces | Part IB, 2007

(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.

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4.II.14A

Metric and Topological Spaces | Part IB, 2007

(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 $M$ .

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1.II.12F

Metric and Topological Spaces | Part IB, 2006

(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.

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2.I.4F

Metric and Topological Spaces | Part IB, 2006

Which 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) $\left\{(x, y, z) \in \mathbf{R}^{3}: z=x^{2}+y^{2}\right\}$ .

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3.I.4F

Metric and Topological Spaces | Part IB, 2006

Which 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 $A$

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4.II.14F

Metric and Topological Spaces | Part IB, 2006

(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.

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1.II.12A

Metric and Topological Spaces | Part IB, 2005

Suppose 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.

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2.I.4A

Metric and Topological Spaces | Part IB, 2005

Let $X$ 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.

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3.I.4A

Metric and Topological Spaces | Part IB, 2005

Show 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 $\operatorname{map}\left(\mathbf{R}, \tau_{1}\right) \rightarrow\left(\mathbf{R}, \tau_{2}\right)$ .

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4.II.14A

Metric and Topological Spaces | Part IB, 2005

Let $(M, d)$ 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.

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