Exercises Attempts

Exercise after §30

1. (a) A $G_\delta$ set in a space $X$ is a set $A$ that equals a countable intersection of open sets of $X$. Show that in a first-countable $T_1$ space, every one-point set is a $G_\delta$ set.

   (b) There is a familiar space in which every one-point set is a $G_\delta$ set, which nevertheless does not satisfy the first countability axiom. What is it?

(a) Let $X$ be a space that is first-countable and $T_1$ and $\{x\}$ a one-point set containing a fixed $x \in X$. Since $X$ is first-countable, there exists a countable collection $\mathcal{B}$ of open sets containing $x$ such that each open set containing $x$ contains at least one of the elements of $\mathcal{B}$.

Note that by Exercise 17.15, for each pair of distinct points of $X$, there exists a neighbourhood of a point not containing the other and vice versa. This implies that if we take pairs of points of $X$ such that one of the points in a pair is $x$, we obtain a family $\{V_\alpha\}_{\alpha \in X \backslash \{x\}}$ of neighbourhoods of $x$ such that $V_\alpha$ does not contain a point $\alpha \in X \backslash \{x\}$. Each of the neighbourhoods $V_\alpha$ also contains at least one of the elements of $\mathcal{B}$; we call it $B_\alpha$. Gathering such $B_\alpha$'s together, we obtain a countable family $\{B_\alpha\}_{\alpha \in X \backslash \{x\}}$ of neighbourhoods of $x$, since it is a subset of $\mathcal{B}$. Taking intersection over such $B_\alpha$'s, we obtain precisely $\{x\}$, because for every point $\alpha \in X \backslash \{x\}$, there exists some $B_\alpha$ such that $\alpha \notin B_\alpha$.

Since the choice of $x \in X$ is arbitrary, it follows that every one-point set is a $G_\delta$ set.

(b) The space $\mathbb{R}^\omega$ equipped with the box topology is not first-countable, but every one-point set in it is a $G_\delta$ set.

To see that a one-point set in $\mathbb{R}^\omega$ is $G_\delta$, given a one-point set $\{\mathbf{x}\}$, where $\mathbf{x}=(x_1,x_2,\ldots,x_n,\ldots)$, consider the following countable collection of open sets (under the box topology):

$$\left\{\prod_{n \in \mathbb{Z}_+}\left(x_n-\frac{1}{i},x_n+\frac{1}{i}\right)\right\}_{i \in \mathbb{Z}_+}.$$

It follows that the intersection of the collection is precisely $\{\mathbf{x}\}$, due to Archimedean property.

To show that $\mathbb{R}^\omega$ in the box topology is not first-countable, let $\mathbf{x}$ be a point in $\mathbb{R}^\omega$ and $\{U_i : i \in \mathbb{Z}_+\}$ a countable collection of neighbourhoods of $\mathbf{x}$. There then exists some basis element $\prod_{n \in \mathbb{Z}_+}(a_n^i,b_n^i)$ that contains $\mathbf{x}$, so that $a_n<x_n<b_n$ for each $n$, and is contained in $U_i$ for each $i$.

Let $c_n=\frac{a_n^n+x_n}{2}$ and $d_n=\frac{x_n+b_n^n}{2}$, then $V=\prod_{n \in \mathbb{Z}_+}(c_k,d_k)$ is a neighbourhood of $\mathbf{x}$ but $U_i \nsubseteq V$ for each $i$.

2. Show that if $X$ has a countable basis $\{B_n\}$, then every basis $\mathcal{C}$ for $X$ contains a countable basis for $X$. [Hint: For every pair of indices $n,m$ for which it is possible, choose $C_{n,m} \in \mathcal{C}$ such that $B_n \subseteq C_{n,m} \subseteq B_m$.]

3. Let $X$ have a countable basis; let $A$ be an uncountable subset of $X$. Show that uncountably many points of $A$ are limit points of $A$.

4. Show that every compact metrizable space $X$ has a countable basis. [Hint: Let $\mathcal{A}_n$ be a finite covering of $X$ by $1/n$-balls.]

5. (a) Show that every metrizable space with a countable dense subset has a countable basis.

   (b) Show that every metrizable Lindelöf space has a countable basis.

9. Let $A$ be a closed subspace of $X$. Show that if $X$ is Lindelöf, then $A$ is Lindelöf.

11. Let $f : X \to Y$ be continuous. Show that if $X$ is Lindelöf, or if $X$ has a countable dense subset, then $f(X)$ satisfies the same condition.

12. Let $f : X \to Y$ be a continuous open map. Show that if $X$ satisfies the first or the second countability axiom, then $f(X)$ satisfies the same axiom.

Exercise after §31

1. Show that if $X$ is regular, every pair of points of $X$ have neighbourhoods whose closures are disjoint.

Suppose that $X$ is regular, then (by the assumption that $X$ is $T_1$) $X$ is also Hausdorff. Therefore, if we let $x,y$ be a pair of (distinct) points in $X$, there exists a neighbourhood $A$ of $x$ and $B$ of $y$ such that $A \cap B = \emptyset$.

Applying Theorem 31.1, there then exists a neighbourhood $U$ of $x$ and $V$ of $y$ such that $\overline{U} \subseteq A$ and $\overline{V} \subseteq B$. Since $A$ and $B$ are disjoint, it follows that $\overline{U}$ and $\overline{V}$ are disjoint as well.

2. Show that if $X$ is normal, every pair of disjoint closed sets have neighbourhoods whose closures are disjoint.

Let $A$ and $B$ be a pair of disjoint closed subsets of $X$. By normality, there exists a neighbourhood $U$ of $A$ and $V$ of $B$ such that $U$ and $V$ are disjoint. By Theorem 31.1, there exists neighbourhoods $U'$ and $V'$ of $A$ and $B$ respectively such that $\overline{U'} \subseteq U$ and $\overline{V'} \subseteq V$. It then follows that $\overline{U'}$ and $\overline{V'}$ are disjoint, as desired.

3. Show that every order topology is regular.

5. Let $f, g : X \to Y$ be continuous; assume that $Y$ is Hausdorff. Show that $\{ x : f(x)=g(x) \}$ is closed in $X$.

8 (reworded). Let $G$ be a compact topological group; let $X$ be a topological space; let $\alpha$ ba an action of $G$ on $X$. If $X$ is Hausdorff, or regular, or normal, or locally compact, or second-countable, so is the orbit space of $\alpha$, denoted by $X/G$.

[Hint: See Exercise 13 of §26.]

Exercise after §32

1. Show that a closed subspace of a normal space is normal.

Let $X$ be a normal space and $Y \subseteq X$ a closed subspace of $X$. Let $A$ and $B$ be disjoint closed subsets of $Y$. By Exercise 17.2 (in Chapter 2), it follows that $A$ and $B$ are also closed in $X$. By normality, we obtain two disjoint open sets $U$ and $V$ in $X$ that contain $A$ and $B$ respectively. It follows that $U \cap Y$ and $V \cap Y$ are two disjoint open sets in $Y$ that contain $A$ and $B$ respectively, thus $Y$ is normal.

2. Show that if $\prod X_\alpha$ is Hausdorff, or regular, or normal, then so is $X_\alpha$. (Assume that each $X_\alpha$ is nonempty.)

3. Show that every locally compact Hausdorff space is regular.

Since $X$ is locally compact and Hausdorff, it follows by Theorem 29.2 in Chapter 3 that for every point $x$ in $X$ and neighbourhood $U$ of $x$, there is a neighbourhood $V$ of $x$ such that $\overline{V}$ is compact and $\overline{V} \subseteq U$. This satisfies Lemma 31.1(a) (although compactness of $\overline{V}$ is not required), so $X$ is regular.

4. Show that every regular Lindelöf space is normal.

6. A space $X$ is said to be completely normal if every subspace of $X$ is normal. Show that $X$ is completely normal if and only if for every pair $A,B$ of separated sets in $X$ (that is, sets such that $\overline{A} \cap B = \emptyset$ and $A \cap \overline{B} = \emptyset$), there exist disjoint open sets containing them. [Hint: If $X$ is completely normal, consider $X \backslash (\overline{A} \cap \overline{B})$].

8. Prove the following:

   Theorem. Every linear continuum $X$ is normal.

   (a) Let $C$ be a nonempty closed subset of $X$. If $U$ is a component of $X \backslash C$, show that $U$ is a set of the form $(c,c')$ or $(c,\infty)$ or $(-\infty,c)$, where $c,c' \in C$.

   (b) Let $A$ and $B$ be closed disjoint subsets of $X$. For each component $W$ of $X \backslash (A \cup B)$ that is an open interval with one endpoint in $A$ and the other in $B$, choose a point $c_W$ of $W$. Show that the set $C$ of the points $c_W$ is closed.

   (c) Show that if $V$ is a component of $X \backslash C$, then $V$ does not intersect both $A$ and $B$.

10. Is every topological group normal?

Exercise after §33

1. Examine the proof of the Urysohn lemma, and show that for given $r$,

   $$f^{-1}(r)=\bigcap_{p>r}U_p-\bigcup_{q<r}U_q,$$

   $p,q$ rational.

2. (a) Show that a connected normal space having more than one point is uncountable.

   (b) Show that a connected regular space having more than one point is uncountable. [Hint: Any countable space is Lindelöf.]

note

In fact, there does exist a connected Hausdorff space that is countably infinite; it is called the irrational slope topology.

3. Give a direct proof of the Urysohn lemma for a metric space $(X,d)$ by setting

$$f(x)=\frac{d(x,A)}{d(x,A)+d(x,B)}.$$

4. Recall that $A$ is a '$G_\delta$ set' in $X$ if $A$ is the intersection of a countable collection of open sets of $X$.

   Theorem. Let $X$ be normal. There exists a continuous function $f : X \to [0,1]$ such that $f(x)=0$ for $x \in A$, and $f(x)>0$ for $x \notin A$, if and only if $A$ is a closed $G_\delta$ set in $X$.

   A function satisfying the requirements of this theorem is said to vanish precisely on $A$.

5. Prove:

   Theorem (Strong form of the Urysohn lemma). Let $X$ be a normal space. There is a continuous function $f : X \to [0,1]$ such that $f(x)=0$ for $x \in A$, and $f(x)=1$ for $x \in B$, and $0<f(x)<1$ otherwise, if and only if $A$ and $B$ are disjoint closed $G_\delta$ sets in $X$.

6. A space $X$ is said to be perfectly normal if $X$ is normal and if every closed set in $X$ is a $G_\delta$ set in $X$.

   (a) Show that every metrizable space is perfectly normal.

   (b) Show that a perfectly normal space is completely normal. For this reason the condition of perfect normality is sometimes called the '$T_6$ axiom.' [Hint: Let $A$ and $B$ be separated sets in $X$. Choose continuous functions $f, g : X \to [0,1]$ that vanish precisely on $\overline{A}$ and $\overline{B}$ respectively. Consider the function $f-g$.]

   (c) There is a familiar space that is completely normal but not perfectly normal. What is it?

7. Show that every locally compact Hausdorff space is completely regular.

8. Let $X$ be completely regular; let $A$ and $B$ be disjoint closed subsets of $X$. Show that if $A$ is compact, there is a continuous function $f : X \to [0,1]$ such that $f(A)=\{0\}$ and $f(B)=\{1\}$.

Exercise after §34

3. Let $X$ be a compact Hausdorff space. Show that $X$ is metrizable if and only if $X$ has a countable basis.

7. A space $X$ is locally metrizable if each point $x$ of $X$ has a neighbourhood that is metrizable in the subspace topology. Show that a compact Hausdorff space $X$ is metrizable if it is locally metrizable. [Hint: Show that $X$ is a finite union of open subspaces, each of which has a countable basis.]

8. Show that a regular Lindelöf space is metrizable if it is locally metrizable. [Hint: A closed subspace of a Lindelöf space is Lindelöf.] Regularity is essential; where do you use it in the proof?

Exercise after §35

1. Show that the Tietze extension theorem implies the Urysohn lemma.

4. Let $Z$ be a topological space. If $Y$ is a subspace of $Z$, we say that $Y$ is a retract of $Z$ if there is a continuous map $r : Z \to Y$ such that $r(y)=y$ for each $y \in Y$.

   (a) Show that if $Z$ is Hausdorff and $Y$ is a retract of $Z$, then $Y$ is closed in $Z$.

   (b) Let $A$ be a two-point set in $\mathbb{R}^2$. Show that $A$ is not a retract of $\mathbb{R}^2$.

   (c) Let $S^1$ be the unit circle in $\mathbb{R}^2$; show that $S^1$ is a retract of $\mathbb{R}^2 \backslash \{\mathbf{0}\}$, where $\mathbf{0}$ is the origin. Can you conjecture whether or not $S^1$ is a retract of $\mathbb{R}^2$?

5. A space $Y$ is said to have the universal extension property if for each triple consisting of a normal space $X$, a closed subset $A$ of $X$, and a continuous function $f : A \to Y$, there exists an extension of $f$ to a continuous map of $X$ into $Y$.

   (a) Show that $\mathbb{R}^J$ has the universal extension property.

   (b) Show that if $Y$ is homeomorphic to a retract of $\mathbb{R}^J$, then $Y$ has the universal extension property.

Exercise after §36

1. Prove that every manifold is regular and hence metrizable. Where do you use the Hausdorff condition?

2. Let $X$ be a compact Hausdorff space. Suppose that for each $x \in X$, there is a neighbourhood $U$ of $x$ and a positive integer $k$ such that $U$ can be embedded in $\mathbb{R}^k$. Show that $X$ can be embedded in $\mathbb{R}^N$for some positive integer $N$.

3. Let $X$ be a Hausdorff space such that each point of $X$ has a neighbourhood that is homeomorphic with an open subset of $\mathbb{R}^m$. Show that if $X$ is compact, then $X$ is an $m$-manifold.

4. An indexed family $\{A_\alpha\}$ of subsets of $X$ is said to be a point-finite indexed family if each $x \in X$ belongs to $A_\alpha$ for only finitely many values of $\alpha$.

   Lemma (The shrinking lemma). Let $X$ be a normal space and $\{U_1,U_2,\ldots\}$ be a point-finite indexed open covering of $X$, then there exists an indexed open covering $\{V_1,V_2,\ldots\}$ of $X$ such that $\overline{V_n} \subseteq U_n$ for each $n$.

An example of a non-Hausdorff manifold.

5. The Hausdorff condition is an essential part of the definition of a manifold; it is not implied by the other parts of the definition. Consider the following space: Let $X$ be the union of the set $\mathbb{R} \backslash \{0\}$ and the two-point set $\{p,q\}$. Topologise $X$ by taking as basis the collection of all open intervals in $\mathbb{R}$ that do not contain $0$, along with all sets of the form $(-a,0) \cup \{p\} (0,a)$ and all sets of the form $(-a,0) \cup \{q\} (0,a)$ for $a > 0$. The space $X$ is called the line with two origins (a.k.a. the bug-eyed line).

   (a) Check that this is a basis for a topology.

   (b) Show that each of the spaces $X \backslash \{p\}$ and $X \backslash \{q\}$ is homeomorphic to $\mathbb{R}$.

   (c) Show that $X$ satisfies the $T_1$ axiom, but is not Hausdorff.

   (d) Show that $X$ satisfies all the conditions for a 1-manifold except for the Hausdorff condition.