§43 Complete Metric Spaces

Ex.43.1

Let X be a metric space.
(a) Suppose that for some $\epsilon>0$, every $\epsilon$-ball in $X$ has compact closure. Show that $X$ is complete.
(b) Suppose that for each $x\in X$ there is an $\epsilon>0$ such that the ball $B(x,\epsilon)$ has compact closure. Show by means of an example that $X$ need not be complete.

§39. Local Finiteness

Ex.39.1

Check the statements in Example 1.

§37. Tychonoff Theorem

Ex.37.1

Let $X$ be a space. Let $\mathscr D$ be a collection of subsets of $X$ that is maximal with respect to the finite intersection property.
(a) Show that $x\in \bar D$ for every $D\in\mathscr D$ iff every open nbhd of $x$ belongs to $\mathscr D$. Which implication uses maximality of $\mathscr D$?
(b) Let $D\in\mathscr D$. Show that if $A\supset D$, then $A\in\mathscr D$
(c) Show that if $X$ satisfies the $T_1$ axiom, there is at most one point belonging to $\bigcap_{D\in\mathscr D}\bar D$

Munkres拓扑是最经典的点集拓扑教材，但奇怪的是网络上只能找到前四章的解答，至于第五到八章，则没有系统完整的解答，只有MSE等论坛上有分散的问答，查起来有些麻烦。我想在空闲时间把自己这学期写的解答放出来，为后来者学习提供方便。考虑到Munkres这本书的受众，解答用英文书写。

我目前只打算写第五到八章的习题解答，第九章之后是代数拓扑的内容，我不会使用这本书来学代数拓扑，大多数人也不会这样做。至于第一到四章，可以参考Vadim的博客，这上面有非常详细的解答，评论区也有适当的讨论或勘误。

对于解答，我再作一些说明：

1. 过于简单的题目我可能会跳过。比如检查正文中的例子的细节、直接的验证定义或对正文定理的直接推广等。
2. Munkres中有些题目是错的，我会在对应的地方注明这点。
3. 我不关注拓扑群，所以拓扑群的习题我都跳过。
4. 一些比较困难的题我可能暂时没做，我会慢慢它们的补充解答。
5. 我说某点的“邻域（nbhd）”一般不特指开邻域，而是指包含此点某开邻域的集合，开的邻域我会单独说明。
6. 解答都是我自己做的或网络收集得到，我不能保证解答的正确性，如果发现有错误之处你可以评论纠错或给我发邮件。

Clarification:

1. I may skip some overly simple problems, such as checking the details of examples in the main text, straightforward verifications of definitions, or direct generalization of theorems from the text.
2. Some problems in Munkres are incorrect; I will note this where applicable.
3. I am not interested in topological groups, so I skip all related exercises.
4. Some challenging problems might remain unfinished for now, and I will gradually add their solutions.
5. When I refer to a “neighborhood (nbhd)” of a point, I do not necessarily mean an open neighborhood; rather, I mean any set containing some open neighborhood of that point. I will specify the openness if needed.
6. All solutions are either my own or collected from online sources. I cannot guarantee their correctness. If you spot any mistakes or typo, feel free to comment to correct me, or send me an email at mcwestlifer@gmail.com.