# Chapter 2

# 1-10

Problem.1 Verify each of the following for arbitrary subsets $A,B$ of a space $X$.

1. $\overline{A\cup B}=\overline A\cup \overline B$;
2. $\overline{A\cap B}\subseteq \overline A\cap \overline B$;
3. $\overline{\overline A}=\overline A$;
4. $(A\cup B)^\circ\supseteq A^\circ \cup B^\circ$;
5. $(A\cap B)^\circ=A^\circ \cap B^\circ$;
6. $(A^\circ)^\circ=A^\circ$;

Show that equality need not hold in (2) and (4).

采用相互包含和定义来证明，仅给出直观。

(1) 只需证明 $A\cup B$ 的极限点属于 $\overline A,\overline B$ 至少其一。

(2) 只需证明 $A,B$ 的公共极限点都属于 $\overline A,\overline B$。

(3) $\overline {\overline A}$ 是包含 $\overline A$ 的最小闭集，而 $\overline A$ 本身是闭集，因此 $\overline {\overline A}=\overline A$。

(4) 注意到 $A^\circ$ 中的邻域也是 $(A\cup B)^\circ$ 中的邻域。

(5) 注意到 $A\cap B$ 的内点必须同时是 $A,B$ 的内点，反过来因为邻域的交仍是邻域，因此 $A,B$ 的内点的交也是 $A\cap B$ 的内点。

(6) 开集的内部是其本身。

(7) 反例都考虑 $A=\mathbb Q,B=\mathbb R\setminus \mathbb Q$。

Problem.2. Find a family of closed subsets of the real line whose union is not closed.

考虑闭区间 $[0,1-1/n]$，则

$$\bigcup _{n=1}^\infty [0,1-1/n]=[0,1)$$

Problem.3. Specify the interior, closure, and frontier of each of the following subsets of the plane:

1. $\{(x,y):1<x^2+y^2\leq 2\}$;
2. $\mathbb E^2$ with both axes removed;
3. $\mathbb E^2-\{(x,\sin(1/x)):x> 0\}$.

(1) 内点为环面 $A(1,2)$，闭包为闭环面 $\overline{A(1,2)}$，边界为两圆 $C_1,C_2$。

(2) 内点为自身，闭包为 $\mathbb E^2$，边界为两坐标轴。

(3) 内点为边界的补，闭包为 $\mathbb E^2$，边界为

$$\{(x,\sin(1/x)):x> 0\}\cup \{(0,y):-1\leq y\leq 1\}$$

Problem.4. Find all the limit points of the following subsets of the real line:

1. $\{(1/m)+(1/n):m,n=1,2,\ldots\}$;
2. $\{(1/n)\sin n:n=1,2,\ldots\}$.

(1) 极限点为 $\{1/n:n=1,2,\ldots\}\cup \{0\}$。

(2) 极限点为 $0$。

Problem.5. If $A$ is a dense subset of a space $X$, and if $O$ is open in $X$, show that $O\subseteq \overline {A\cap O}$.

对于任意 $x\in O$，因为 $A$ 在 $X$ 中稠密，所以 $x$ 的任一邻域 $x\in U\subseteq O$ 都满足

$$(U\setminus \{x\})\cap A\neq \varnothing$$

这说明

$$(U\setminus\{x\}\cap (A\cap O)\neq \varnothing$$

Problem.6. If $Y$ is a subspace of $X$, and $Z$ a subspace of $Y$, prove that $Z$ is a subspace of $X$.

$Z$ 上的开集为形如 $Z\cap U$ 的集合，其中 $U$ 是 $Y$ 的开集；$Y$ 上的开集为形如 $Y\cap V$ 的集合，其中 $V$ 是 $X$ 的开集。因此，$Z$ 上的开集为

$$Z\cap U=Z\cap (Y\cap V)=(Z\cap Y)\cap V=Z\cap V$$

其中 $V$ 是 $X$ 的开集，因此 $Z$ 是 $X$ 的子空间。

Problem.7. Suppose $Y$ is a subspace of $X$. Show that a subset of $Y$ is closed in $Y$ if it is the intersection of $Y$ with a closed set in $X$. If $A$ is a subset of $Y$, show that we get the same answer whether we take the closure of $A$ in $Y$, or intersect $Y$ with the closure of $A$ in $X$.

如果 $A$ 是 $Y$ 的闭集，则存在 $X$ 的开集 $U$ 使得

$$Y\setminus A=Y\cap U$$

取关于 $Y$ 的补集，得到

$$K=Y\setminus(Y\setminus K)=Y\setminus U=Y\cap U^c$$

如果取 $A$ 在 $Y$ 中的闭包 $\overline A^Y$，则 $\overline A^Y\subseteq Y\cap \overline A^X$。而 $\overline A^X$ 是 $X$ 中包含 $A$ 的最小闭集，因此只能有 $\overline A^Y=Y\cap \overline A^X$。

Problem.8. Let $Y$ be a subspace of $X$. Given $A\subseteq Y$, write $\mathring A_Y$ for the interior of $A$ in $Y$ and $\mathring A_X$ for the interior of $A$ in $X$. Prove that $\mathring A_X\subseteq \mathring A_Y$, and give an example to show the two may not be equal.

$X$ 中的开集限制在 $Y$ 的子空间拓扑下还是开集。考虑 $X=\mathbb R,Y=\mathbb Q$，则 $A=(0,1)\cap \mathbb Q$，在 $Y$ 中的内部为自身，而在 $X$ 中的内部为空集。

Problem.9. Let $Y$ be a subspace of $X$. If $A$ is open(closed) in $Y$, and if $Y$ is open(closed) in $X$, show that $A$ is open(closed) in $X$.

$A$ 是 $Y$ 的开集，则存在 $Y$ 的开集 $U$ 使得

$$A=Y\cap U$$

这表明 $A$ 可以写成两个 $X$ 的开集的交，因此 $A$ 是 $X$ 的开集。闭集同理，参考 [Arm.C2,T7] 的结论。

Problem.10. Show that the frontier of a set always contains the frontier of its interior. How does the frontier of $A\cup B$ relate to the frontiers of $A$ and $B$?

边界点的定义是不属于内点和外点的点，因此 $\mathring A$ 的边界点不是 $\mathring A$ 的内点，即不是的点，所以不是 $A$ 的内点；另一方面，它不是 $\mathring A$ 的外点，因此对于其任意邻域，都包含 $\mathring A$ 中的点，即包含 $A$ 中的点，因此不是 $A$ 的外点。综上，$\partial A\supseteq \partial (\mathring A)$。

# 11-20

Problem.11. Let $X$ be the set of real numbers and $\beta$ the family of all subsets of the form $\{x:a\leq x<b,a<b\}$. Prove that $\beta$ is a base for a topology on $X$ and that in this topology each member of $\beta$ is both open and closed. Show that this topology does not have a countable base.

首先 $\beta$ 至少满足

$$\bigcup [n,n+1)=\mathbb R,\quad [n,n+1)\cap [n+1,n+2)=\varnothing$$

在这个拓扑结构下，$(-\infty,a),[b,+\infty)$ 都是开集，因此任意基础开集

$$[a,b)=\mathbb R\setminus \bigl((-\infty,a)\cup [b,+\infty)\bigr)$$

都是闭集。

证明该拓扑没有可数基，反证法，设 $B_1,B_2,\ldots$ 是一个可数基 $\mathcal B'$，那么任意开集都可以由 $\mathcal B'$ 中的成员的并表示。考虑开集 $[x,x+1)$，则一定存在 $B_i\in\mathcal B'$ 使得

$$x\in B_i\subseteq [x,x+1)$$

否则，$x$ 没有包含在 $\mathcal B'$ 中的任何一个成员里，矛盾。这说明 $\inf B_i=x$。$x$ 可以是任意实数，因此 $\mathcal B'$ 至少有不可数多个成员，矛盾。

Problem.12. Show that if $X$ has a countable base for its topology, then $X$ contains a countable dense subset. A space whose topology has a countable base is called a second countable (第二可数) space. A space which contains a countable dense subset is said to be separable (可分).

根据依赖选择公理，从每个基础开集中选取一个点，构成可数集 $A$。对于任意 $x\in X\setminus A$ 及其邻域 $U$，因为 $\mathcal B$ 是 $X$ 的基，所以存在 $B\in \mathcal B$ 使得 $x\in B\subseteq U$，而 $B$ 中有 $A$ 的点，因此 $A\cap (U\setminus \{x\})\neq \varnothing$。这说明 $A$ 在 $X$ 中稠密。

Problem.13. If $f:\mathbb R\to\mathbb R$ is a map(i.e., a continuous function), show that the set of points which are left fixed by $f$ is a closed subset of $\mathbb R$. If $g$ is a continuous real-valued function on $X$ show that the set $\{x:g(x)=0\}$ is closed.

显然映射 $g:=f-\mathrm {id}:\mathbb R\to\mathbb R$ 是连续的，所以只需要考虑 $g$ 的零点集，因为零点的极限点仍然是零点，所以零点集是闭集。

Problem.14. Prove that the function $h(x)=e^x/(1+e^x)$ is a homeomorphism from the real line to the open interval $(0,1)$.

需要验证双射、连续及其逆连续。双射由单调性推出，连续性由复合推出，逆连续性同样由复合推出，且

$$h^{-1}(y)=\ln\biggl(\frac y{1-y}\biggr)$$

Problem.15. Let $f:\mathbb E^1\to \mathbb E^1$ be a map and define its graph $\Gamma_f:\mathbb E^1\to\mathbb E^2$ by $\Gamma_f(x)=(x,f(x))$. Show that $\Gamma_f$ is continuous and that its image(taken with the topology induced from $\mathbb E^2$) is homeomorphic to $\mathbb E^1$.

由连续函数的性质，$f(B_{\delta(\varepsilon)}(x))\subseteq B_\varepsilon(f(x))$，其中不妨设 $\delta(\varepsilon)<\varepsilon$，则

$$(\Gamma_f)(B_{\delta(\varepsilon)}(x))\subseteq B_{\sqrt 2\varepsilon}(\Gamma_f(x)),\quad \forall x\in X,\varepsilon>0$$

这说明 $\Gamma_f$ 在 $\Gamma_f(\mathbb E^1)$ 的任意基础开集的原像在 $\mathbb E^1$ 中是开集，因而 $\Gamma_f$ 是连续的。显然 $\Gamma_f$ 是双射（考虑第一分量），而且其逆映射 $\pi:\Gamma_f(\mathbb E^1)\to \mathbb E^1$ 也是连续的

$$\pi\circ B_{\delta(\varepsilon)}(\Gamma_f(x))\subseteq B_\varepsilon(x),\quad \forall x\in X,\varepsilon>0$$

因此 $\Gamma_f$ 是一个同胚。

Problem.16. What topology must $X$ have if every real-valued function defined on $X$ is continuous?

考虑测试函数

$$f:X\to\mathbb R,\quad f(x)=\begin{cases}0,&x\neq x_0\\1,&x=x_0\end{cases}$$

则 $\{x_0\}=f^{-1}(B(1,1/2))$ 是 $X$ 的开集，所以 $X$ 必须是离散拓扑。

Problem.17. Let $X$ denote the set of all real numbers with the finite-complement topology, and define $f:\mathbb E^1\to X$ by $f(x)=x$. Show that $f$ is continuous, but is not a homeomorphism.

在有限补拓扑下，开集都是补集有限的集合，这些集合都是 $\mathbb E^1$ 的开集；但反过来不成立。

Problem.18. Suppose $X=A_1\cup A_2\cup \ldots$, where $A_n\subseteq \mathring A_{n+1}$ for each $n$. If $X\to Y$ is a function such that, for each $n$, $f|_{A_n}:A_n\to Y$ is continuous with respect to the induced topology on $A_n$, show that $f$ is itself continuous.

对于任意开集 $V\subseteq Y$，结合相对开集的定义，有

$$f^{-1}(V)=\bigcup _{n=1}^\infty f|_{A_n}^{-1}(V)=\bigcup^\infty_{n=1}A_n\cap U_n=\bigcup^\infty_{n=1}A_n\cap \bigcup^\infty_{n=1}U_n=\bigcup^\infty_{n=1}U_n$$

这是因为 $U_n\cap A_n=U_{n+1}\cap A_n$。

Problem.19. The characteristic function (特征函数) of a subset $A$ of a space $X$ is the real-valued function on $X$ which assigns the value $1$ to points of $A$ and $0$ to all other points. Describe the frontier of $A$ in terms of this function.

边界点是特征函数不连续的点。

Problem.20. An open map (开映射) is one which sends open sets to open sets; a closed map (闭映射) takes closed sets to closed sets. Which of the following maps are open or closed?
(a) The exponential map $x\mapsto e^{ix}$ from the real line to the circle.
(b) The folding map $f:\mathbb R^2\to \mathbb R^2$ given by $f(x,y)=(x,|y|)$.
(c) The map which winds the plane three times on itself given, in terms of complex numbers, by $z\mapsto z^3$.

(a) 开映射，但不是闭映射。

(b) 闭映射，但不是开映射。

(c) 开映射，闭映射。

# 21-26

Problem.21. Show that the unit ball (单位球) (the set of points whose coordinates satisfy $x_1^2+x_2^2+\ldots +x_n^2\leq 1$) and the unit cube (单位立方体) (points whose coordinates satisfy $|x_i|\leq 1,1\leq i\leq n$) are homeomorphic if they are both given the subspace topology from $\mathbb R^n$.

考虑连续映射 $f:\mathbb S^n\to \partial \mathrm{Cube}$，定义为

$$f:(x_1,\ldots,x_n)\mapsto \frac{(x_1,\ldots,x_n)}{\max(|x_1|,\ldots,|x_n|)}$$

从而可以诱导单位球和单位立方体的连续映射，这个连续映射只改变模长，不改变方向，因此是双射。容易验证逆映射（只关心球面和立方体面，单位球和单位立方体的映射是自然的）也是连续的。

$$g:(y_1,\ldots,y_n)\mapsto \frac{(y_1,\ldots,y_n)}{\sqrt{y_1^2+\ldots+y_n^2}}$$

Problem.22. Find a Peano curve which fills out the unit square in $\mathbb R^2$.

Problem.23.

Problem.24.

Problem.25.

Problem.26.

Problem.27. Show $d(x,A)=0$ iff $x$ is a point of $\overline A$.

根据度量空间关于极限点的定义即可证明。

Problem.28. If $A,B$ are disjoint closed subsets of a metric space, find disjoint open sets $U,V$ such that $A\subseteq U$ and $B\subseteq V$.

由于两个闭集不交，因此对于任意 $a\in A$，都有 $d(a,B)>0$，取

$$U=\bigcup_{a\in A}B\bigl(a,d(a,B)/2\bigr),\quad V=\bigcup_{b\in B}B\bigl(b,d(b,A)/2\bigr)$$

则 $U,V$ 是所求的开集。

Problem.29. Show one can define a distance function on an arbitrary set $X$ by $d(x,y)=1$ if $x\neq y$ and $d(x,x)=0$. What topology does $d$ give to $X$?

离散拓扑，每个点都是开集。

Problem.30. Show that every closed subset of a metric space is the intersection of a countable number of open sets.

考虑开集族

$$U_n=\{x:d(x,A)<1/n\},\quad n=1,2,\ldots$$

# 31-36

Problem.31. If $A,B$ are subsets of a metric space, their 分离距离 (distance apart) $d(A,B)$ is the infimum of the numbers $d(x,y)$ where $x\in A$ and $y\in B$. Find two disjoint closed subsets of the plane which are zero distance apart. The diameter (直径) of $A$ is the supremum of the numbers $d(x,y)$ where $x,y\in A$. Check that both of the closed sets which you have just found have infinite diameter.

考虑闭集，满足分离距离为 $0$：

$$A=\{(x,1/x):x>0\},B=\{(x,-1/x):x>0\}$$

并且直径均为无穷大。

Problem.32. If $A$ is a closed subset of a metric space $X$, show that any map $f:A\to\mathbb R^n$ can be extended over $X$.

根据 Tietze 延拓定理可知，这只不过是同时考虑 $n$ 个实值函数的延拓问题。

Problem.33. Find a map from $\mathbb R^1\setminus \{0\}$ to $\mathbb R^1$ which cannot be extended over $\mathbb R^1$.

考虑映射 $x\mapsto 1/x$。

Problem.34. Let $f:C\to C$ be the identity map of the unit circle in the plane. Extend $f$ to a map from $\mathbb R^2\setminus \{0\}$ to $C$. Would you expect to be able to extend $f$ over all of $\mathbb R^2$?

定义

$$\tilde f:\mathbb R^2\setminus \{0\}\to \mathbb R,\quad x\mapsto f(x/|x|)$$

Problem.35. Given a map $f:X\to\mathbb R^{n+1}\setminus\{0\}$ find a map $g:X\to \mathbb S^n$ which agrees with $f$ on the set $f^{-1}(\mathbb S^n)$.

定义

$$g:X\to \mathbb S^n,\quad x\mapsto {f(x)}/{|f(x)|}$$

Problem.36. If $X$ is a metric space and $A$ closed in $X$, show that a map $f:A\to\mathbb S^n$ can always be extended over a neighborhood of $A$, in other words over a subset of $X$ which is a neighborhood of each point of $A$. (Think of $\mathbb S^n$ as a subspace of $\mathbb R^{n+1}$ and extend $f$ to a map of $X$ into $\mathbb R^{n+1}$. Now use Problem 35.)

考虑 Problem 32，可以将 $A$ 延拓定义到 $\mathbb R^{n+1}$；只要邻域足够小，映射值就不会是 $0$，因此可以使用 Problem 35 中的方法进行延拓。

# Chapter 3

待完成：Problem 5

# 1-10

Problem.1. Find an open cover of $\mathbb R^1$ which does not contain a finite subcover. Do the same for $[0,1)$ and $(0,1)$.

对于 $\mathbb R^1$，考虑开覆盖 $(n,n+2)$；对于 $[0,1)$，考虑开覆盖 $[0,1-1/n)$；对于 $(0,1)$，考虑开覆盖 $(1/n,1-1/n)$。

Problem.2. Let $S\supseteq S_1\supseteq S_2\supseteq \ldots$ be a nested sequence of squares in the plane whose diameters tend to zero as we proceed along the sequence. Prove that the intersection of all these squares consists of exactly one point.

反证法，不然存在两点 $x,y$ 都在所有的正方形中，则 $d(x,y)>0$，但随着正方形的直径趋于 $0$，总存在一个正方形使得其直径小于 $d(x,y)$，矛盾。

Problem.3. Use the Heine-Borel theorem to show that an infinite subset of a closed interval must have a limit point.

闭区间是紧空间，如果无限子集没有极限点，则考虑其每个点的邻域，它们不包含其他该无限子集的点，这些邻域构成一个开覆盖，但没有有限子覆盖，矛盾。

Problem.4. Rephrase the definition of compactness in terms of closed sets.

紧空间当且仅当对于任意闭集族，如果任意有限子集的交非空，则整个闭集族的交非空。这是对开覆盖定义的对偶表述。

Problem.5. Which of the following are compact?
(a) The space of rational numbers;
(b) $\mathbb S^n$ with a finite number of points removed;
(c) The torus with an open disc removed;
(d) The Klein bottle;
(e) The Mobius strip with its boundary circlce removed.

(a) 不是；(b) 不是；(c) 是；(d) 是；(e) 不是。

这些特殊曲面可以嵌入到 $\mathbb R^n$ 中。

Problem.6. Show that the Hausdorff condition cannot be relaxed in the theorem below: a one-one, onto, and continuous function from a compact space $X$ to a Hausdorff space $Y$ is a homeomorphism.

只需要寻找紧集非闭的例子，例如有限补拓扑。

Problem.7. Show that Lebesgue's lemma fails for the plane.

Lebesgue 引理要求紧空间。基于此，我们构造平面的开覆盖

$$U_0=(-\infty,1)\times \mathbb R,\quad U_n=(n-1-1/n,n+1/n)\times \mathbb R$$

Problem.8. (Lindelof's theorem). If $X$ has a countable base for its topology, prove that any open cover of $X$ contains a countable subcover.

设可数基的基础开集为 $B_n$，任意开覆盖为 $U_\alpha$，则对于任意 $x\in X$，存在 $U_\alpha$ 使得 $x\in U_\alpha$，由可数基的定义，存在 $B_n$ 使得 $x\in B_n\subseteq U_\alpha$。记 $U_\alpha:= U_{\alpha(n)}$，则

$$X=\bigcup_{n=1}^\infty B_n\subseteq \bigcup_{n=1}^\infty U_{\alpha(n)}\subseteq \bigcup_{\alpha}U_\alpha$$

Problem.9. Prove that two disjoint compact subsets of a Hausdorff space always possess disjoint neighborhoods.

记不相交紧集为 $A,B$，先固定 $a\in A$，考虑对任意 $b\in B$，使用 Hausdorff 性质，再使用$B$ 的紧性选取有限开覆盖，得到 $a$ 的邻域 $U_a$ 和 $B$ 的邻域 $V_a$ 使得 $U_a\cap V_a=\varnothing$。变动 $a\in A$，使用 $A$ 的紧性，选取有限开覆盖。

Problem.10. Let $A$ be a compact subset of a metric space $X$. Show that the diameter of $A$ is equal to $d(x,y)$ for some pair of points $x,y\in A$. Given $x\in X$, show that $d(x,A)=d(x,y)$ for some $y\in A$. Given a closed subset $B$, disjoint from $A$, show that $d(A,B)>0$.

由下确界定义构造 $d(x_n,y_n)\to \mathrm{diam}(A)$ 的序列，使用紧性选取收敛子列，得到极限点 $x,y$，则 $d(x,y)=\mathrm{diam}(A)$。对 $d(x,A)$ 同理。对于 $d(A,B)$，假设 $d(A,B)=0$，则由上存在 $a\in A$ 使得 $d(a,B)=0$，这说明 $a$ 是 $B$ 的极限点，矛盾。

# 11-20

Problem.11. Find a topological space and a compact subset whose closure is not compact.

$\mathbb R$ 配备射线拓扑，开集形如 $(-\infty,a)$，则 $\{0\}$ 是紧集，但其闭包 $[0,+\infty)$ 不是紧集。

Problem.12. Do the real numbers with the finite-complement topology form a compact space? Answer the same question for the half-open interval topology (see Problem 11 of Chapter 2).

有限补拓扑是紧空间，特别地，其子空间都是紧空间。半开拓扑不是紧空间，例子为

$$[0,1)\subseteq \bigcup_{n=1}^\infty [0,1-1/n)$$

Problem.13. Let $f:X\to Y$ be a closed map with the property that the inverse image of each point of $Y$ is a compact subset of $X$. Show that $f^{-1}(K)$ is compact whenever $K$ is a compact in $Y$. Can you remove the condition that $f$ be closed?

考虑 $f^{-1}(K)$ 的任意闭集族，有限交非空，因为 $f$ 是闭映射，将这些闭集映到 $Y$ 中，成为 $K$ 的闭集族，使用 $K$ 的紧性，闭集族的交非空，拉回 $f^{-1}(K)$ 可知它的闭集族的交也非空。

Problem.14. If $f:X\to Y$ is a one-one map, and if $f:X\to f(X)$ is a homeomorphism, when we give $f(X)$ the induced topology from $Y$, we call $f$ an embedding (嵌入) of $X$ in $Y$. Show that a one-one map from a compact space to a Hausdorff space must be an embedding.

对于紧空间的任意闭集 $U$，$f(U)$ 是闭集，结合双射推出逆映射连续。

Problem.15. A space is locally compact (局部紧) if each of its points has a compact neighborhood. Show that the following are all locally compact: any compact space; $\mathbb R^n$; any discrete space; any closed subset of a locally compact space. Show that the space of rationals is not locally compact. Check that local compactness is preserved by a homeomorphism.

紧邻域就是该点在其内部的紧集。对于紧空间，取全空间；对于 $\mathbb R^n$，取闭球是紧集；对于离散拓扑，取单点集，因为单点集是开集；对于局部紧空间的闭子集，存在紧邻域，取其与闭子集的交，在子空间拓扑下是紧邻域。有理数空间不是局部紧的，因为在子空间拓扑下，任意内部非空的紧集都包含无理数点，据此构造反例。

同胚保持局部紧性质，因为其保持开集和紧集，以及序关系。

Problem.16. Suppose $X$ is locally compact and Hausdorff. Given $x\in X$ and a neighborhood $U$ of $x$, find a compact neighborhood of $x$ which is contained in $U$.

对于任意点 $x$ 都存在紧邻域 $V$，取 $V\setminus U$，这是闭集，从而是紧集。由 Hausdorff 性质，可以证明存在不交开集 $S,T$ 使得

$$x\in S\subseteq V\setminus T\subseteq U,\quad V\setminus U\subseteq T$$

$V\setminus T$ 是 $x$ 的紧邻域，且包含在 $U$ 中。

Problem.17. Let $X$ be a locally compact Hausdorff space which is not compact. Form a new space by adding one extra point, usually denoted by $\infty$, to $X$ and taking the open sets of $X\cup\{\infty\}$ to be those of $X$ together with sets of the form $(X-K)\cup\{\infty\}$, where $K$ is a compact subset of $X$. Check the axioms for a topology, and show that $X\cup \{\infty\}$ is a compact Hausdorff space which contains $X$ as a dense subset. The space $X\cup \{\infty\}$ is called the one-point compactification (一点紧化) of $X$.

验证拓扑公理，进行形式运算即可，注意紧集的有限并仍然是紧集。根据 Hausdorff 性质，紧集都是闭集，因此 $X\cup \{\infty\}$ 的任意开覆盖都至少包含一个形如 $(X-K)\cup\{\infty\}$ 的开集，剩下的紧集 $K$ 有有限子覆盖，所以 $X\cup \{\infty\}$ 是紧空间。Hausdorff 空间是显然的。将 $X$ 视为稠密子空间也是显然的，因为有形如 $(X-K)\cup\{\infty\}$ 的开集。

Problem.18. Prove that $\mathbb R^n\cup\{\infty\}$ is homeomorphic to $\mathbb S^n$. (Think first of the case $n=2$. Stereographic projection gives a homeomorphism between $\mathbb R^2$ and $\mathbb S^2$ minus the north pole, points 'out towards infinity' in the plane becoming points near to the north pole on the sphere. Think of replacing the north pole in $\mathbb S^2$ as adding a point at $\infty$ to $\mathbb R^2$.)

考虑球极投影，记北极点为 $N$，补充定义 $f(N)=\infty$。则

$$\mathbb S^n\setminus \{N\}\cong \mathbb R^n$$

事实上，由 Problem.17 的一点紧化，只需要验证 $\mathbb S^n$ 包含 $N$ 的开集与 $\mathbb R^n\cup \{\infty\}$ 包含 $\infty$ 的开集一一对应即可。

Problem.19. Let $X$ and $Y$ be locally compact Hausdorff spaces and let $f:X\to Y$ be an onto map. Show $f$ extends to a map from $X\cup\{\infty\}$ onto $Y\cup\{\infty\}$ iff $f^{-1}(K)$ is compact for each compact subset $K$ of $Y$ (i.e., iff $f$ is a proper map (真映射)). Deduce that if $X$ and $Y$ are homeomorphic spaces then so are their one-point compactifications. Find two spaces which are not homeomorphic but which have homeomorphic one-point compactifications.

充分性验证连续性，只需要验证包含 $\infty$ 的开集的原像也是开集。必要性，考虑 $(Y-K)\cup\{\infty\}$ 的原像 $(X-T)\cup \{\infty\}$，这要求 $T$ 是紧集，而正好 $f^{-1}(K)=T$。由连续函数将紧集映到紧集，所以同胚空间的一点紧化也是同胚的。对于不同胚的空间，其一点紧化也可以是同胚的。考虑 $X=[0,1),Y=(0,1)$ 配备子空间拓扑，它们不同胚，同样可以通过连通性推出矛盾，因为 $X$ 去掉 $0$ 还是连通的，但是 $Y$ 中每一个点都是分割点。但它们的一点紧化都是 $\mathbb S^1$。补充一点，$X$ 和 $\mathbb S^1$ 也不同胚，因为连通性推出矛盾。

Problem.20. If $X\times Y$ has the product topology, and if $A\subseteq X,B\subseteq Y$, show that

$$\overline {A\times B}=\overline A\times \overline B,\quad (A\times B)^\circ=A^\circ \times B^\circ$$

$$\partial (A\times B)=(\partial A\times \overline B)\cup (\overline A\times \partial B)$$

一切从基础开集出发即可，注意到乘积拓扑将基础开集的乘积作为基础开集。证明前两个结论后，注意到最后一个结论是它们的差集。

# 21-30

Problem.21. If $A$ and $B$ are compact, and if $W$ is a neighbourhood of $A\times B$ in $X\times Y$, find a neighbourhood $U$ of $A$ in $X$ and a neighbourhood $V$ of $B$ in $Y$ such that $U\times V\subseteq W$.

固定 $a\in A$，$\{a\}\times B$ 是紧集，因此考虑开覆盖，得到有限个基础开集 $U_{a_i}\times V_{a_i}$ 覆盖，再取子开集 $O_a:=\bigcap U_{a_i}\cup \bigcup V_{a_i}$，变动 $a\in A$，使用 $A$ 的紧性，可以得到有限个 $O_{a_j}$ 覆盖 $A\times B$，同样 $O_a$ 的构造方法得到所求的邻域。

Problem.22. Prove that the product of two second-countable spaces is second-countable, and that the product of two separable spaces is separable.

第二可数空间指的是有可数基的拓扑空间。乘积空间的拓扑基是两个空间的拓扑基的乘积，所以是可数的，因此乘积空间是第二可数空间。可分空间是包含可数稠密子集的空间。根据可数的性质，乘积仍然是可数的。

Problem.23. Prove that $[0,1)\times [0,1)$ is homeomorphic to $[0,1]\times [0,1)$.

直观上看，前者是正方形去掉上、右边界（包含端点），后者是正方形去掉右边界（包含端点）。参照 Chapter 2 Problem 21 的方法，可以构造正方形和圆盘的同胚，只不过前者去掉了一条闭半圆弧，后者去掉了四分之一的闭半圆弧，构造角度同胚调整即可。

Problem.24. Let $x_0\in X$ and $y_0\in Y$. Prove that the function $f:X\to X\times Y, g:Y\to X\times Y$ defined by $f(x)=(x,y_0)$ and $g(y)=(x_0,y)$ are embeddings (as defined in Problem 14).

只需考虑 $f$ 的情形。$f$ 是双射，通过如下复合，注意到连续映射的乘积仍然是连续映射

$$X\xrightarrow{(\mathrm{id},c_{y_0})}X\times \{y_0\}\xrightarrow{\iota}X\times Y$$

常值映射、恒等映射和嵌入映射都是连续的。同样可以证明逆映射（投影）是连续的。

Problem.25. Show that the diagonal map $\Delta:X\to X\times X$ defined by $\Delta(x)=(x,x)$ is indeed a map, and check that $X$ is Hausdorff iff $\Delta(X)$ is closed in $X\times X$.

同上，这是恒等映射的乘积，再复合上嵌入映射，所以是连续映射。充分性，因为 $\Delta(X)$ 是闭集，所以对于任意不同两点 $x,y$，$(x,y)$ 都不是 $\Delta(X)$ 的极限点，因此存在基础开集 $U\times V$ 使得 $(x,y)\in U\times V$ 且 $U\cap V=\varnothing$。必要性，直接由 Hausdorff 性质选取不交开集，乘积开集内部不包含对角线上的点。

Problem.26. We know that the projections $p_1:X\times Y\to X$, $p_2:X\times Y\to Y$ are open maps. Are they always closed?

因为乘积空间的拓扑结构是使得投影映射连续的最小拓扑结构，同时投影映射将基础开集映为开集，所以是开映射。但不一定是闭映射，考虑 $\mathbb R\times \mathbb R$ 的投影映射，将闭集 $\{xy=1:x,y>0\}$ 映为开集 $(0,\infty)$。

Problem.27. Given a countable number of spaces $X_1,X_2,\ldots$, a typical point of the product $\prod X_i$ will be written $x=(x_1,x_2,\ldots)$. The product topology (乘积拓扑) on $\prod X_i$ is the smallest topology for which all of the projections $p_i:\prod X_i\to X_i$, $p_i(x)=x_i$ are continuous. Construct a base for this topology from the open sets of the spaces $X_1,X_2,\ldots$.

这是最小拓扑，因此至少要求包含 $p_i^{-1}(U_i)$，其中 $U_i$ 是 $X_i$ 的开集。由此生成的拓扑即为乘积拓扑，形如

$$\prod_{i=1}^n p_i^{-1}(U_i)=U_1\times U_2\times \ldots \times U_n\times \prod_{i=n+1}^\infty X_i$$

这里的下标是对应任意 $n$ 个空间，并不一定是前 $n$ 个空间。

Problem.28. If each $X_i$ is a metric space, the topology on $X_i$ being induced by a metric $d_i$, prove that

$$d(x,y)=\sum_{i=1}^\infty \frac{1}{2^i}\frac{d_i(x_i,y_i)}{1+d_i(x_i,y_i)}$$

defines a metric on $\prod X_i$ which induces the product topology.

验证度量的良定性和度量公理，显然。

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Problem.29. The box topology (箱拓扑) on $\prod X_i$ has as base all sets of the form $U_1\times U_2\times \ldots$ where $U_i$ is open in $X_i$. Show that the box topology contains the product topology, and that the two are equal iff $X_i$ is an indiscrete space for all but finitely many values of $i$. ($X$ is an indiscrete space (不可分空间) if the only open sets are $\varnothing$ and $X$.)

、、、、、、、、、、、、、、、、、、、、、、、、、、、、、、、、

Problem.30. Let $X$ be the set of all points in the plane which have at least one rational coordinate. Show that $X$, with the induced topology, is a connected space.

考虑任意两点，取有理分量构造折线段即可。

# 31-40

Problem.31. Give the set of real numbers the finite-complemenet topology. What are the components of the resulting space? Answer the same question for the half-open interval topology.

对于有限补空间，只有一个连通分支。假设全空间不连通，则存在既开又闭的非平凡子集，矛盾。对于半开区间拓扑，首先全空间不连通，因为其可以拆解成 $(-\infty,a)\cup[a,\infty)$ 为两个开集的不交并。进一步可以看到 $[a,b)$ 也不连通，所以只能有单点集是连通分支。

Problem.32. If $X$ has only a finite number of components, show that each component is both open and closed. Find a space none of whose components are open sets.

连通分支构成 $X$ 的划分，闭集的有限并仍然是闭集。考虑 Problem 31 的半开拓扑，则其连通分支都是单点集，但单点集不是开集。

Problem.33. (Intermediate value theorem). If $f:[a,b]\to\mathbb R^1$ is a map such that $f(a)<0$ and $f(b)>0$, use the connectedness of $[a,b]$ to establish the existence of a point $c$ for which $f(c)=0$.

连续映射保持连通性。

Problem.34. A space $X$ is locally connected (局部连通) if for each $x\in X$, and each neighbourhood $U$ of $x$, there is a connected neighbourhood $V$ of $x$ which is contained in $U$. Show that any euclidean space, and therefore any space which is locally euclidean (like a surface), is locally connected. If $X=\{0\}\cup\{1/n:n=1,2,\ldots\}$ with the subspace topology from the real line, show that $X$ is not locally connected.

Euclid 空间是局部连通的，因为任意开球都是连通的。同胚保持连通性，因此局部 Euclid 空间也是局部连通的。对于 $X$，在 $0$ 处不是局部连通的。

Problem.35. Show that local connectedness is preserved by a homeomorphism, but need not be preserved by a continuous function.

这正是 Problem 34 要证明的问题。因为同胚保持开集和连通性，如果 $A$ 是局部连通的，并且 $A\cong B$，那么对于任意 $x\in B$ 及其邻域 $U$，取 $f^{-1}(U)=V\subseteq A$，则 $f^{-1}(x)\in V$ 且由局部连通性知道存在连通邻域 $W\subseteq V$，因此 $f(W)\subseteq U$ 是 $x$ 的连通邻域。

Problem.36. Show that $X$ is locally connected iff every component of each open subset of $X$ is an open set.

必要性。如果 $X$ 局部连通，则对于任意开子集 $U$ 作为子空间，其任一连通分支 $C$ 上取任意一点 $x\in C$，存在连通邻域 $V$，则 $V\cap C\neq \varnothing$，所以由连通分支定义知道，$V\subseteq C$，因此 $C$ 是开集。充分性。对于任意点 $x$ 及其邻域 $U$，则考虑 $U$ 的子空间拓扑，其上某一条连通分支 $C$ 包含 $x$，则 $C\subseteq U$ 是开集，且是 $x$ 的连通邻域。

Problem.37. Show that the continuous image of a path-connected space is path-connected.

连续像指的是一个连续函数作用在这个空间上得到的像空间。对于像空间的任意两点 $x,y$，考虑原像空间的两点（不一定唯一）使得 $f(x_0)=x,f(y_0)=y$，由路径连通性，存在路径连接 $x_0,y_0$，将该路径映射到像空间即得到所求路径。

Problem.38. Show that $\mathbb S^n$ is path-connected for $n>0$.

对于球面任意两点，选取第三点作为极点，使用球极投影将两点映射到平面上，连接后再映射回球面即可。

Problem.39. Prove that product of two path-connected spaces is path-connected.

对于乘积空间的任意两点 $(x_1,y_1),(x_2,y_2)$，根据分量空间的道路连通性，可以选取道路，期间第二分量保持 $y_1$ 不变，这样就可以连接 $(x_1,y_1)$ 和 $(x_2,y_1)$；再选取道路，对第二分量同样操作即可。

Problem.40. If $A$ and $B$ are path-connected subsets of a space, and if $A\cap B$ is nonempty, prove that $A\cup B$ is path-connected.

选取 $A\cap B$ 中的点 $x_0$，对于任意两点 $a\in A,b\in B$，连接 $a$ 和 $x_0$，再连接 $x_0$ 和 $b$ 即可。

# 41-44

Problem.41. Find a path-connected subset of a space whose closure is not path-connected.

拓扑学家的正弦曲线。补充一点，连通集的闭包是连通的，但道路连通集的闭包不一定是连通的。

$$\{(x,\sin(1/x)):0<x\leq 1\}\cup \{(0,y):-1\leq y\leq 1\}$$

对于 $(0,1)$，它和正弦曲线上的点 —— 不妨设为 $p$—— 不是道路连通的，因为连接它们的道路 $\gamma(t)$ 是连续映射，当道路参数 $t\to 1$ 时，道路点应趋近于 $\gamma(t)\to (0,1)$，但在正弦曲线上不论 $t$ 多小，道路点的纵坐标都不会趋近于 $1$（横坐标迫使它依照正弦曲线向 $y$ 轴移动）。矛盾。

Problem.42. Show that any indiscrete space is path-connected.

不可分空间是平凡的。所以取道路 $\gamma(t)$ 在 $[0,1)$ 上恒等于某个点，在 $1$ 处取另一个点。在不可分空间中，这是连续映射，所以确实是道路。

Problem.43. A space $X$ is locally path-connected if for each $x\in X$, and each neighbourhood $U$ of $x$, there is a path-connected neighbourhood $V$ of $x$ which is contained in $U$. Is the topologist's sine curve locally path-connected? Convert the space $\{0\}\cup \{1/n:n=1,2,\ldots\}$ into a subspace of the plane which is path-connected but not locally path-connected.