Exercises Attempts

Exercise after §23

1. Let $\mathcal{T}$ and $\mathcal{T}'$ be two topologies on $X$. If $\mathcal{T}' \supseteq \mathcal{T}$, what does connectedness of $X$ in one topology imply about connectedness in the other?

Suppose that $X$ is connected under $\mathcal{T}'$, then the only subsets of $X$ that are both open and closed in $X$ under $\mathcal{T}'$ are the empty set and $X$ itself. Since the open sets in $\mathcal{T}$ are contained in $\mathcal{T}'$, it follows that the only open and closed subsets of $X$ under $\mathcal{T}$ are also the empty set and $X$ itself, implying that $X$ is connected in $\mathcal{T}$.

If $X$ is connected under $\mathcal{T}$, it is not in general true that $X$ is connected under $\mathcal{T}'$, as there may be other open and closed subsets of $X$ under $\mathcal{T}'$. For example, $\mathbb{R}$ is connected under the indiscrete topology, but not connected under the discrete topology.

2. Let $\{A_n\}$ be a sequence of connected subspaces of $X$, such that $A_n \cap A_{n+1} \neq \emptyset$ for all $n$. Show that $\bigcup A_n$ is connected.

3. Let $\{A_\alpha\}$ be a collection of connected subspaces of $X$; let $A$ be a connected subspace of $X$. Show that if $A \cap A_\alpha \neq \emptyset$ for all $\alpha$, then $A \cup (\bigcup A_\alpha)$ is connected.

4. Show that if $X$ is an infinite set, it is connected in the finite complement topology.

5. A space is totally disconnected if its only connected subspaces are one-point sets. Show that if $X$ has the discrete topology, then $X$ is totally disconnected. Does the converse hold?

9. Let $A$ be a proper subset of $X$, and let $B$ be a proper subset of $Y$. If $X$ and $Y$ are connected, show that

$$(X \times Y) \backslash (A \times B)$$

is connected.

10. Let $\{X_\alpha\}_{\alpha \in J}$ be an indexed family of connected spaces; let $X$ be the product space

    $$X=\prod_{\alpha \in J} X_\alpha.$$

    Let $\mathbf{a}=(a_\alpha)$ be a fixed point of $X$.

    (a) Given any finite subset $K$ of $J$, let $X_K$ denote the subspace of $X$ consisting of all points $\mathbf{x}=(x_\alpha)$ such that $x_\alpha=a_\alpha$ for $\alpha \in K$. Show that $X_K$ is connected.

    (b) Show that the union $Y$ of the spaces $X_K$ is connected.

    (c) Show that $X$ equals the closure of $Y$; conclude that $X$ is connected.

12. Let $Y \subseteq X$; let $X$ and $Y$ be connected. Show that if $A$ and $B$ form a separation of $X \backslash Y$, then $Y \cup A$ and $Y \cup B$ are connected.

Exercise after §24

1. (a) Show that no two of the spaces $(0,1)$, $(0,1]$, and $[0,1]$ are homeomorphic. [Hint: What happens if you remove a point from each of these spaces?]

   (b) Suppose that there exists embeddings $f : X \to Y$ and $g : Y \to X$. Show by means of an example that $X$ and $Y$ need not be homeomorphic.

   (c) Show $\mathbb{R}^n$ and $\mathbb{R}$ are not homeomorphic if $n>1$.

2. Let $f : S^1 \to \mathbb{R}$ be a continuous map. Show there exists a point $x$ of $S^1$ such that $f(x)=f(-x)$.

note

Recall that $S^1$ is defined as the space of all complex numbers $z$ for which $|z|=1$.

3. Let $f : X \to X$ be continuous. Show that if $X = [0,1]$, there is a point $x$ such that $f(x)=x$. The point $x$ is called a fixed point of $f$. What happens if $X$ equals $[0,1)$ or $(0,1)$?

4. Let $X$ be an ordered set in the order topology. Show that if $X$ is connected, then $X$ is a linear continuum.

10. Show that if $U$ is an open connected subspace of $\mathbb{R}^2$, then $U$ is path connected. [Hint: Show that given $x_0 \in U$, the set of points that can be joined to $x_0$ by a path in $U$ is both open and closed in $U$.]

Exercise after §25

4. Let $X$ be locally path connected. Show that every connected open set in $X$ is path connected.

5. Let $X$ denote the rational poitns of the interval $[0,1] \times \{0\}$ of $\mathbb{R}^2$. Let $T$ denote the union of all line segments joining the point $p = (0,1)$ to points of $X$.

   (a) Show that $T$ is path connected, but is locally connected only at the point $p$. (b) Find a subset of $\mathbb{R}^2$ that is path connected but is locally connected at none of its points.

6. A space $X$ is said to be weakly locally connected at $x$ if for every neighbourhood $U$ of $x$, there is a connected subspace of $X$ contained in $U$ that contains a neighbourhood of $x$. Show that if $X$ is weakly locally connected at each of its points, then $X$ is locally connected. [Hint: Show that components of open sets are open.]

9. Let $G$ be a topological group; let $C$ be the component of $G$ containing the identity element $e$. Show that $C$ is a normal subgroup of $G$. [Hint: If $x \in G$, then $xC$ is the component of $G$ containing $x$.]

10. Let $X$ be a space. Let us define $x \sim y$ if there is no separation $X = A \cup B$ of $X$ into disjoint open sets such that $x \in A$ and $y \in B$.

    (a) Show this relation is an equivalence relation. The equivalence classes are called the quasicomponents of $X$. (b) Show that each component of $X$ lies in a quasicomponent of $X$, and that the components and quasicomponents of $X$ are the same if $X$ is locally connected. (c) Let $K$ denote the set $\left\{\frac{1}{n} : n \in \mathbb{Z}_+\right\}$, and let $-K$ denote the set $\left\{-\frac{1}{n} : n \in \mathbb{Z}_+\right\}$. Determine the components, path components, and quasicomponents of the following subspaces of $\mathbb{R}^2$:

    $$A = (K \times [0,1]) \cup \{(0,0)\} \cup \{(0,1)\}$$ $$B = A \cup ([0,1] \times \{0\})$$ $$C = (K \times [0,1]) \cup (-K \times [-1,0]) \cup ([0,1] \times -K) \cup ([-1,0] \times K)$$

Exercise after §26

1. (a) Let $\mathcal{T}$ and $\mathcal{T}'$ be two topologies on the set $X$; suppose that $\mathcal{T}' \supseteq \mathcal{T}$. What does compactness of $X$ under one of these topologies imply about the compactness under the other?

   (b) Show that if $X$ is compact Hausdorff under both $\mathcal{T}$ and $\mathcal{T}'$, then either $\mathcal{T}$ and $\mathcal{T}'$ are equal or they are not comparable.

3. Show that a finite union of compact subspaces of $X$ is compact.

4. Show that every compact subspace of a metric space is bounded in that metric and is closed. Find a metric space in which not every closed bounded subspace is compact.

5. Let $A$ and $B$ be disjoint compact subspaces of the Hausdorff space $X$. Show that there exists disjoint open sets $U$ and $V$ containing $A$ and $B$, respectively.

6. Show that if $f : X \to Y$ is continuous, where $X$ is compact and $Y$ is Hausdorff, then $f$ is a closed map (that is, $f$ carries closed sets to closed sets).

7. Show that if $Y$ is compact, then the projection $\pi_1 : X \times Y \to X$ is a closed map.

8 (reworded). Prove that for a function $f : X \to Y$ where $Y$ is compact Hausdorff, we have that $f$ is continuous if and only if the graph of $f$,

$$G_f := \{ (x ,f(x)) : x \in X\}$$

is closed in $X \times Y$.

12. Let $p : X \to Y$ be a closed continuous surjective map such that $p^{-1}(\{y\})$ is compact, for each $y \in Y$ (Such a map is called a perfect map.) Show that if $Y$ is compact, then $X$ is compact.

    [Hint: If $U$ is an open set containing $p^{-1}(\{y\})$, there is a neighbourhood $W$ of $y$ such that $p^{-1}(W)$ is contained in $U$.]

13. Let $G$ be a topological group.

    (a) Let $A$ and $B$ be subspaces of $G$. If $A$ is closed and $B$ is compact, show that $A \cdot B$ is closed. [Hint: If $c$ is not in $A \cdot B$, find a neighbourhood $W$ of $c$ such that $W \cdot B^{-1}$ is disjoint from $A$.]

    (b) Let $H$ be a subgroup of $G$; let $p : G \to G/H$ be the quotient map. If $H$ is compact, show that $p$ is a closed map.

    (c) Let $H$ be a compact subgroup of $G$. Show that if $G/H$ is compact, then $G$ is compact.

Exercise after §27

1. Prove that if $X$ is an ordered set in which every closed interval is compact, then $X$ has the least upper bound property.

note

This is the converse of Theorem 27.1.

2. Let $X$ be a metric space with metric $d$; let $A \subseteq X$ be nonempty.

   (a) Show that $d(x,A)=0$ if and only if $x \in \overline{A}$.

   (b) Show that if $A$ is compact, $d(x,A) = d(x,a)$ for some $a \in A$.

   (c) Define the _$\varepsilon$-neighbourhood _ of $A$ in $X$ to be the set

   $$U(A,\varepsilon)=\{x : d(x,A)<\varepsilon\}.$$

   Show that $U(A,\varepsilon)$ equals the union of the open balls $B_d(a,\varepsilon)$ for $a \in A$.

   (d) Assume that $A$ is compact; let $U$ be an open set containing $A$. Show that some $\varepsilon$-neighbourhood of $A$ is contained in $U$.

   (e) Show the result in (d) need not hold if $A$ is closed but not compact.

4. Show that a connected metric space having more than one point is uncountable.

5. Let $X$ be a compact Hausdorff space; let $\{A_n\}$ be a countable collection of closed sets of $X$. Show that if each set $A_n$ has empty interior in $X$, then the union $\bigcup A_n$ has empty interior in $X$. [Hint: Imitate the proof of Theorem 27.7.]

note

This is a special case of the Baire category theorem.

6. Let $A_0$ be the closed interval $[0,1]$ in $\mathbb{R}$. Let $A_1$ be the set obtained from $A_0$ by deleting its 'middle third' $\left(\frac{1}{3},\frac{2}{3}\right)$. Let $A_2$ be the set obtained from $A_1$ by deleting its 'middle thirds' $\left(\frac{1}{9},\frac{2}{9}\right)$ and $\left(\frac{7}{9},\frac{8}{9}\right)$. In general, define $A_n$ by the equation

   $$A_n = A_{n-1} \backslash \bigcup_{k=0}^{\infty}\left(\frac{1+3k}{3^n},\frac{2+3k}{3^n}\right).$$

   The intersection

   $$C=\bigcap_{n \in \mathbb{Z}_+}A_n$$

   is called the Cantor set; it is a subspace of $[0,1]$.

   (a) Show that $C$ is totally disconnected.

   (b) Show that $C$ is compact.

   (c) Show that each set $A_n$ is a union of finitely many disjoint closed intervals of length $1/3^n$; and show that the end points of these intervals lie in $C$.

   (d) Show that $C$ has no isolated points.

   (e) Conclude that $C$ is uncountable.

Exercise after §28

3. Let $X$ be limit point compact.

   (a) If $f : X \to Y$ is continuous, does it follow that $f(X)$ is limit point compact?

   (b) If $A$ is a closed subset of $X$, does it follow that $A$ is limit point compact?

   (c) If $X$ is a subspace of the Hausdorff space $Z$, does it follow that $X$ is closed in $Z$?

   note

   It is not true in general that the product of two limit point compact spaces is limit point compact, even if the Hausdorff condition is assumed. However, the (counter)examples are remarked to be sophisticated, such as the Novak space.

4. A space $X$ is said to be countably compact if every countable open covering of $X$ contains a finite subcollection that covers $X$. Show that for a $T_1$ space $X$, countable compactness is equivalent to limit point compactness. [Hint: If no finite subcollection of $U_n$ covers $X$, choose $x_n \notin U_1 \cup \cdots \cup U_n$, for each $n$.]

5. Show that $X$ is countably compact if and only if every nested sequence $C_1 \supseteq C_2 \supseteq \cdots$ of closed nonempty sets of $X$ has a nonempty intersection.

6. Let $(X,d)$ be a metric space. If $f : X \to X$ satisfies the condition

$$d(f(x),f(y))=d(x,y)$$

for all $x,y \in X$, then $f$ is called an isometry of $X$. Show that if $f$ is an isometry and $X$ is compact, then $f$ is bijective and hence a homeomorphism. [Hint: If $a \in f(X)$, choose $\varepsilon$ so that the $\varepsilon$-neighbourhood of $a$ is disjoint from $f(X)$. Set $x_1=a$, and $x_{n+1}=f(x_n)$ in general. Show that $d(x_n,x_m) \geqslant \varepsilon$ for $n \neq m$.]

7. Let $(X,d)$ be a metric space. If $f$ satisfies the condition

$$d(f(x),f(y)) < d(x,y)$$

for all $x,y \in X$ with $x \neq y$, then $f$ is called a shrinking map. If there is a number $\alpha<1$ such that

$$d(f(x),f(y)) \leqslant \alpha d(x,y)$$

for all $x,y \in X$, then $f$ is called a contraction. A fixed point of $f$ is a point $x$ such that $f(x)=x$.

(a) If $f$ is a contraction and $X$ is compact, show $f$ has a unique fixed point. [Hint: Define $f^1=f$ and $f^{n+1}=f \circ f^n$. Consider the intersection $A$ of the sets $A_n = f^n(X)$.]

(b) Show more generally that if $f$ is a shrinking map and $X$ is compact, then $f$ has a unique fixed point. [Hint: Let $A$ be as before. Given $x \in A$, choose $x_n$ so that $x = f^{n+1}(x_n)$. If $a$ is the limit of some subsequence of the sequence $y_n=f^n(x_n)$, show that $a \in A$ and $f(a)=x$. Conclude that $A=f(A)$, so that $\mathrm{diam}\; A=0$.]

(c) Let $X=[0,1]$. Show that $f(x)=x-x^2/2$ maps $X$ into $X$ and is a shrinking map that is not a contraction. [Hint: Use the mean-value theorem of calculus.]

(d) The result in (a) holds if $X$ is a complete metric space, such as $\mathbb{R}$; see the exercises of §43. The result in (b) does not: Show that the map $f : \mathbb{R} \to \mathbb{R}$ given by $f(x)=[x+(x^2+1)^{1/2}]/2$ is a shrinking map that is not a contraction and has no fixed point.

Exercise after §29

1. Show that the rationals $\mathbb{Q}$ are not locally compact.

3. Let $X$ be a locally compact space. If $f : X \to Y$ is continuous, does it follow that $f(X)$ is locally compact? What if $f$ is both continuous and open? Justify your answer.

5. If $f : X_1 \to X_2$ is a homeomorphism of locally comapct Hausdorff spaces, show $f$ extends to a homeomorphism of their one-point compactifications.

6. Show that the one-point compactification of $\mathbb{R}$ is homeomorphic with the circle $S^1$.

8. Show that the one-point compactification of $\mathbb{Z}_+$ is homeomorphic with the subspace $\{0\} \cup \{\frac{1}{n} : n \in \mathbb{Z}_+ \}$ of $\mathbb{R}$.

9. Show that if $G$ is a locally compact topological group and $H$ is a subgroup, then $G/H$ is locally compact.

10. Show that if $X$ is a Hausdorff space that is locally compact at the point $x$, then for each neighbourhood $U$ of $x$, there is a neighbourhood $V$ of $x$ such that $\overline{V}$ is compact and $\overline{V} \subseteq U$.