Supplementary exercises

A collection $\mathcal{B} = \left\{ B_i : i \in I\right\}$ of subsets of $X$ is called a basis and its elements are called basis elements if the following properties are satisfied:

Show that the set of balls $\left\{ B(x, r), x \in X, r > 0\right\}$ is a basis for the topology of a metric space $X$; that is, show that it is a basis and the topology it determines is the metric topology.

Show that any open set in $T_{\mathcal{B}}$ (the topology determined by a basis $\mathcal{B}$) is a union of basis elements.

Suppose the topology for $Y$ is determined by a basis. Show that $f : X \to Y$ is continuous iff, for each basis element $B$ of $Y$, $f^{-1}(B)$ is open.

Show that the open intervals give a basis for the topology of $\mathbb {R}$.

Combine the last two exercises to show that a map to $\mathbb {R}$ is continuous iff, for each open interval $I \subset \mathbb {R}$, we have $f^{-1}(I)$ is open. Formulate and prove an analogous statement for maps to a metric space in terms of balls.

Let $X$ be a metric space with metric $d$ and let $A$ be a subset of $X$. Show that the metric topology on $A$ given by $d$ is the same as the subspace topology.

Suppose $A \subset B \subset X$, and $B$ has the subspace topology. Show that, if $A$ is open in $B$ and $B$ is open in $X$, then $A$ is open in $X$.

If $x, y \in \mathbb {R}^{2}$, let $d(x, y) = \left|x_1 - y_1\right|$. Which of the three properties of a metric does $d$ satisfy?

On $\mathbb {R}^{2} \setminus \left\{ 0\right\}$, write each point in polar coordinates as $(r, \theta )$, where $0 \leq \theta < 2\pi$. Define $d((r_1, \theta_1), (r_2, \theta_2)) = \left|r_1 - r_2\right| + \left|\theta_1 - \theta_2\right|$. Show that this gives a metric on $\mathbb {R}^{2} \setminus \left\{ 0\right\}$ but the topology formed is not the usual topology.

A point $x$ is called a limit point of a set $A$ if every open set $U$ containing $x$ intersects $A \setminus \left\{ x\right\}$ in a nonempty set; that is, $U \cap (A \setminus \left\{ x\right\} ) \neq \emptyset$ for $U$ open, $x \in U$. Denote the limit points of $A$ by $A'$.

Show that a set $A$ is closed iff it contains all of its limit points.

Find the limit points of the following subsets of $\mathbb {R}$:

Let $X$ be a metric space and $A \subset X$. Show that $x$ is a limit point of $A$ iff every ball $B(x, r)$ contains infinitely many points of $A$.

Show that, in a Hausdorff space $X$ with subset $A$, $x$ is a limit point of $A$ iff every open set containing $x$ contains infinitely many points of $A$.

Show that $A$ is closed iff $A = \overline{A}$, and $A$ is open iff $A = {A}^{\mathrm{o}}$.

Show that $\overline{A} = A \cup A'$, where $A'$ denotes the set of limit points of $A$.

A topological space $X$ is called limit point compact if every infinite set has a limit point.

Show that a compact space is limit point compact.

A topological space $X$ is called limit point compact if every infinite set has a limit point.

Show that if $X$ is a metric space, then $X$ is limit point compact iff it is sequentially compact iff it is compact.

Show that if $f : X \to Y$ is continuous at $x$ and $x_n$ is a sequence converging to $x$, then $f(x_n)$ converges to $f(x)$.

Show that a map $f : (X, d) \to (Y, d')$ between metric spaces is continuous at $x$ iff for every sequence $x_n$ which converges to $x$, the sequence $f(x_n)$ converges to $f(x)$.

Show that, in a Hausdorff space, the limit of a sequence is well defined; that is, if $x_n$ converges to $x$ and to $y$, then $x = y$.

Show that, in a Hausdorff space, if $x_n$ converges to $x$, then $x$ is the only limit point of the set of all of values $\left\{ x_n : n \in \mathbb {N}\right\}$. Is the converse true? Give a proof or counterexample.

Show that a finite set in a Hausdorff space is closed.

A Hausdorff space is called regular if, given $x \in X$ and a closed set $C$ with $x \notin C$, then there are disjoint open sets $U$ and $V$ with $x \in U$ and $C \subset V$.

Show that a compact Hausdorff space is regular.

Suppose $x \in U \subset X$, where $X$ is regular and $U$ is open. Show that there is an open set $V$ with $x \in V \subset \overline{V} \subset U$.

A Hausdorff space is called normal if, whenever $C, D$ are disjoint closed subsets, then there are disjoint open sets $U, V$ with $C \subset U$, $D \subset V$.

Show that a compact Hausdorff space is normal.

Suppose $C \subset U \subset X$, where $X$ is normal, $C$ is closed and $U$ is open. Then show that there is an open set $V$ with $C \subset V \subset \overline{V} \subset U$.

Show that a metric space is normal.

Let $C$ be a subset of a metric space $X$. For each $x \in X$, define the distance from $x$ to $C$ by $d(x, C) = \inf \left\{ d(x, y) : y \in C\right\}$.

Show that $\left\{ x : d(x, C) = 0\right\} = \overline{C}$. Show that if $C$ is closed and $x \notin C$, then $d(x, C) > 0$.

Let $C$ be a subset of a metric space $X$, and $d(x, C) = \inf \left\{ d(x, y) : y \in C\right\}$ the distance from $x \in X$ to $C$.

Show that $C_\epsilon = \left\{ y : d(y, C) < \epsilon \right\}$, where $\epsilon > 0$, is an open set containing $C$.

Let $C$ be a subset of a metric space $X$, and $d(x, C) = \inf \left\{ d(x, y) : y \in C\right\}$ the distance from $x \in X$ to $C$.

Show that the function $f : X \to \mathbb {R}$ given by $f(x) = d(x, C)$ is continuous.

Suppose $C, D$ are disjoint closed sets in a metric space $X$. Use the notion of the distance from a point to a set (i.e. $d(x, C) = \inf \left\{ d(x,y) : y \in C\right\}$) to define $d(C, D)$ in terms of $d(x, D)$ for $x \in C$. Give an example to show that this distance could be $0$. Show that if $X$ is compact, then the distance must be positive.

Suppose $(X, d)$ is a metric space and $A, B$ are disjoint closed sets. Show that the function

is a continuous real-valued function $f : X \to [-1, 1]$ with $f^{-1}\left\{ -1\right\} = A$, $f^{-1}\left\{ 1\right\} = B$. The existence of a function from $X$ to $[-1, 1]$ which is $-1$ on $A$ and $1$ on $B$ is called Urysohn's Lemma and holds in the more general situation of a normal space.

Suppose $X$ is a metric space and $C \subset X$ is a closed subset, with a continuous function $f : C \to \mathbb {R}$. This exercise leads through a proof that there is a continuous extension $F : X \to \mathbb {R}$. This result is called the Tietze extension theorem and holds in the more general situation of a normal space.

Urysohn's lemma states that for a normal space $X$ with disjoint closed sets $A, B$, there is a continuous function $f : X \to [-1, 1]$ with $A \subset f^{-1}\left\{ -1\right\}$ and $B \subset f^{-1}\left\{ 1\right\}$. The Tietze extension theorem states that for a normal space $X$ with a closed subset $C$ and a continuous function $f : C \to \mathbb {R}$, there is a continuous function $g : X \to \mathbb {R}$ with $f(x) = g(x)$ for $x \in C$.

Show that Urysohn's lemma is equivalent to the Tietze extension theorem.

Show that $\mathbb {Q}$ is dense in $\mathbb {R}$, and hence $\mathbb {R}$ is separable (i.e. $\mathbb {R}$ has a countable dense subset).

A space $X$ is called first countable if, for each $x \in X$, there is a countable basis of open sets containing $x$; that is, there is a collection $\left\{ B_n : n \in \mathbb {N}\right\}$ of open sets containing $x$ so that, if $U$ is an open set containing $x$, then there is a set $B_k \subset U$. This is called a neighborhood basis.

Show that a metric space is first countable.

A space $X$ is called first countable if, for each $x \in X$, there is a countable basis of open sets $\left\{ B_n : n \in \mathbb {N}\right\}$ containing $x$ so that, if $U$ is an open set containing $x$, then there is a set $B_k \subset U$.

Show that, if $X$ is first countable and $x$ is a limit point of $C$, then there is a sequence $x_i \in C$ which converges to $x$.

A space $X$ is called first countable if, for each $x \in X$, there is a countable basis of open sets $\left\{ B_n : n \in \mathbb {N}\right\}$ containing $x$ so that, if $U$ is an open set containing $x$, then there is a set $B_k \subset U$.

Show that if $X$ is first countable, Hausdorff, and compact, then $X$ is sequentially compact.

A space $X$ is called second countable if there is a countable basis for the topology of $X$. A metric space is called separable if there is a countable set $\left\{ x_n : n \in \mathbb {N}\right\}$ so that every open set contains at least one $x_n$ (we say $\left\{ x_n\right\}$ is dense in $X$ and $\left\{ x_n\right\}$ is a countable dense subset).

Show that a metric space is separable iff it is second countable.

A space $X$ is called countably compact if every countable open cover has a finite subcover. A space $X$ is called second countable if there is a countable basis for the topology of $X$.

Show that compactness implies countable compactness, and that the converse holds in a second countable space.

Consider the space $X = A \cup B$, where $A = \left\{ (x, \sin 1/x) : 0 < x \leq 1\right\}$ and $B = \left\{ (0, y) : -1 \leq y \leq 1\right\}$. The space $X$ is called the topologist's sine curve. This and the next two exercises show that $X$ is connected but not path connected.

The topologist’s sine curve—two views.

Show that $A \cup B = \overline{A}$.

Consider the space $X = A \cup B$, where $A = \left\{ (x, \sin 1/x) : 0 < x \leq 1\right\}$ and $B = \left\{ (0, y) : -1 \leq y \leq 1\right\}$ (the topologist's sine curve; see the figure in Exercise 1.9.44).

Show that the closure of a connected set is connected, thus implying that $A \cup B$ is connected.

Consider the space $X = A \cup B$, where $A = \left\{ (x, \sin 1/x) : 0 < x \leq 1\right\}$ and $B = \left\{ (0, y) : -1 \leq y \leq 1\right\}$ (the topologist's sine curve; see the figure in Exercise 1.9.44). We show here that $A \cup B$ is not path connected. Suppose $A \cup B$ were path connected. Let $f : [0, 1] \to A \cup B$ be a path with $f(0) = (0, 0)$ and $f(1) = (1, \sin 1)$. Consider the set $S = \left\{ t : f([0, t]) \subset B\right\}$. Let $p : \mathbb {R}^{2} \to \mathbb {R}$ be $p(x, y) = x$.

A topological space $X$ is called locally path connected at $x$ if for each open set $V$ containing $x$, there is a path connected open set $U$ with $x \in U \subset V$. It is called locally path connected if it is locally path connected at each $x \in X$. It is called locally connected at $x$ if, for each open set $V$ containing $x$, there is a connected open set $U$ with $x \in U \subset V$. It is called locally connected if it is locally connected for each $x \in X$.

Show that a locally path connected space is locally connected.

Show that the topologist's sine curve (see the figure in Exercise 1.9.44) is not locally connected, where a space is locally connected at $x$ if for each open set $V$ containing $x$, there is a connected open set $U$ with $x \in U \subset V$.

Show that an open set in $\mathbb {R}^{n}$ is locally path connected, where a space is locally path connected at $x$ if for each open set $V$ containing $x$, there is a path connected open set $U$ with $x \in U \subset V$.

Show that the path components of a locally path connected space are open sets, where a space is locally path connected at $x$ if for each open set $V$ containing $x$, there is a path connected open set $U$ with $x \in U \subset V$.

Show that if $X$ is locally path connected and connected, then $X$ is path connected.

Give an example of a path connected space which is not locally path connected.

Define an equivalence relation on $X$ by $x \sim y$ if there is a connected set containing both $x$ and $y$. The equivalence classes are called the components of $X$.

A collection $\mathcal{D}$ of subsets of $X$ is said to satisfy the finite intersection property (F.I.P.) if for every finite subcollection $\left\{ D_1, \ldots , D_k\right\}$ of $\mathcal{D}$, the intersection $D_1 \cap \cdots \cap D_k \neq \emptyset$. Show that $X$ is compact iff for every collection $\mathcal{D}$ of closed sets satisfying the F.I.P., the intersection of all of the elements of $\mathcal{D}$ is nonempty.

Let $(X, d)$ be a compact metric space, and $f : X \to X$ continuous. $x \in X$ is called a fixed point of $f$ if $f(x) = x$. $f$ is called a contraction if there is a number $a < 1$ such that $d(f(x), f(y)) \leq a \, d(x, y)$ for all $x, y \in X$. Show that a contraction has a unique fixed point. This result is known as the contraction mapping principle and plays a key role in analysis.

$X$ is called locally compact at $x$ if there is an open set $U$ and a compact set $C$ with $x \in U \subset C$. It is called locally compact if it is locally compact at each $x \in X$.

Show that a compact space is locally compact.

Show that $\mathbb {R}^{n}$ is locally compact, where a space is locally compact at $x$ if there is an open set $U$ and a compact set $C$ with $x \in U \subset C$.

Show that if $X$ is Hausdorff and locally compact at $x$, then there is an open set $U$ containing $x$ so that $\overline{U}$ is compact.

Suppose $X$ is a locally compact Hausdorff space. Show that if $x \in U \subset X$, $U$ open, then there exists an open set $V$ containing $x$ such that $\overline{V}$ is compact and $\overline{V} \subset U$.

Suppose that $X$ is a locally compact Hausdorff space. Form a new space $X^+$, called the one-point compactification of $X$, as $X^+ = X \cup \left\{ p\right\}$, the disjoint union of $X$ and an added point $p$. A set $U \subset X^+$ is called open if (1) $U$ is an open set in $X$, or (2) $p \in U$ and $X^+ \setminus U$ is a compact set in $X$.

Show that if $X$ is a locally compact Hausdorff space and $Y$ is a compact Hausdorff space with $Y \setminus \left\{ y_0\right\} \simeq X$, then $Y \simeq X^+$, where $X^+$ denotes the one-point compactification of $X$.

Show that, if $X$ is compact, the one-point compactification of $X$ is $X^+ = X \sqcup \left\{ p\right\}$, where the set $\left\{ p\right\}$ is an open set in the disjoint union.

Consider the space $\mathbb {R}^{\infty }$, which is the product of a countably infinite number of copies of $\mathbb {R}$. This is given the product topology with basis the sets which are a product of a finite number of intervals with a product of copies of the reals:

Show that this space is not locally compact.

Show that $\infty$ (the figure-eight curve) is not homeomorphic to O (the circle).

The next four exercises concern the homeomorphism type of the letters of the alphabet. In each case, assume the letter is written in the sans serif style, with no adornments: $\mathrm{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$.

Show that the letter X is not homeomorphic to the letter Y, but that the letter Y is homeomorphic to the letter T.

The letters of the alphabet are written in the sans serif style, with no adornments: $\mathrm{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$.

Construct a homeomorphism between the letter D and the letter O.

The letters of the alphabet are written in the sans serif style, with no adornments: $\mathrm{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$.

Prove that the letter A is not homeomorphic to the letter B.

The letters of the alphabet are written in the sans serif style, with no adornments: $\mathrm{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$.

Group the letters of the alphabet into equivalence classes so that equivalent letters are homeomorphic and nonequivalent letters are not homeomorphic.

Prove that if an open set $U \subset \mathbb {R}^{2}$ is path connected, then any two points in $U$ can actually be connected by a polygonal path in $U$.

For each of the following subsets of $\mathbb {R}^{2}$ indicate which of the following properties it possesses, namely, (i) compact; (ii) connected; (iii) path connected; (iv) open; (v) closed:

Show that the torus is not homeomorphic to an open set in $\mathbb {R}^{2}$.

Show that if $A \subset S^1$ with $A \neq S^1$, then $A$ is not homeomorphic to $S^1$.

Put an equivalence relation $\sim$ on $\mathbb {R}^{2}$ by saying that two points are equivalent if they both lie on the circle of radius $r$ about the origin. Show that $\mathbb {R}^{2} / \sim$, with the quotient topology, is homeomorphic to $[0, \infty )$.

Identify all points in the lower hemisphere of the sphere $S^2$. Show that the resulting quotient space $S^2 / \sim$ is homeomorphic to $S^2$.

Identify points on the boundary circles of an annulus $A$ between the circles of radius 1 and radius 2 that lie on the same ray from the origin. Show that the resulting quotient space $A / \sim$ is homeomorphic to the torus $T^2$.

Identify the points on the outer circle of an annulus $A$ to one point and the points on the inner circle to a (different) point. Show that $A / \sim$ is homeomorphic to $S^2$. Describe what space you would get if you identified the points on both circles to a single point.

Consider the quotient space $X$ formed from two copies of $D^1 \times D^1$ using $f : \left\{ -1, 1\right\} \times D^1 \to \left\{ -1, 1\right\} \times D^1$ by $f(-1, x) = (1, -x)$, $f(1, y) = (-1, y)$, $X = (D^1 \times D^1) \cup_f (D^1 \times D^1)$. Decide whether $X$ is homeomorphic to the annulus or the Möbius band, and prove your assertion.

Show that the upper hemisphere of $S^2$ is homeomorphic to $D^2$ and similarly for the lower hemisphere. Use this to show $S^2$ is homeomorphic to $D^2 \cup_g D^2$ for $g : S^1 \to S^1$ and determine $g$.

Show that $D^2 \cup_f D^2$, where $f : K \to K$ is $f(x) = x$, and $K = \left\{ (x_1, x_2) \in S^1 : x_1 \geq 0\right\}$ is homeomorphic to $D^2$.