习题 22, § 2

令$X$为具有可数基$(U_n)$的豪斯多夫拓扑空间，并令$R$为$X$上的豪斯多夫等价关系，使得$X/R$的每个点具有可数基数系统邻域。 证明$X/R$的拓扑有可数的基。 [Let $\varphi : X \to X/R$ be the canonical mapping, and show that the interiors of finite unions of sets $\varphi(U_n)$ form a base of the topology of $X/R$: if $V$ is a neighbourhood of a point $z \in X/R$, and $(W_k)$ is a sequence of sets belonging to the base $(U_n)$, contained in $\overline{\varphi^{-1}(V)}$ and covering $\overline{\varphi^{-1}(z)}$, show that there is a finite number of indices $k$ such that the union of the $\overline{\varphi^{-1}(W_k)}$ is a neighbourhood of $z$. Argue by contradiction: if this statement were false we could construct a sequence $(\gamma_n)$ of distinct points of $X/R$, tending to $z$ and such that $\gamma_n$ did not belong to the union of the $\varphi(W_k)$ for $k \leq n$; then show that the union of the $\overline{\varphi^{-1}(\gamma_n)}$ would be closed in $X$.]

证明如果假设 $R$ 是闭集且每个等价类 mod $R$ 都是紧致的（类似方法），那么同样的结论是有效的。