Section 30: Problem 18 Solution

Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.

James R. Munkres

Let $G$ be a first-countable topological group. Show that if $G$ has a countable dense subset, or is Lindelof, then $G$ has a countable basis. [Hint: Let $\{B_{n}\}$ be a countable basis at e. If $D$ is a countable dense subset of $G$ , show the sets $dB_{n}$ , for $d\in D$ , form a basis for $G$ . If $G$ is Lindelof, choose for each n a countable set $С_{n}$ such that the sets $cB_{n}$ , for $с\in C_{n}$ , cover $G$ . Show that as $n$ ranges over $\mathbb{Z}_{+}$ , these sets form a basis for $G$ .]

Occasionally we use some results from the Supplementary Exercises of Chapter 2. Please make sure you have good knowledge and understanding of those facts.

Suppose $\{B_{n}\}$ is a countable basis at $e$ .

First, we suppose that $G$ has a countable dense subset $D$ , and show that $\{d\cdot B_{n}|d\in D\}$ form a (countable) basis. For an arbitrary $x\in G$ and its neighborhood $U$ , take a symmetric neighborhood $V$ of $e$ such that $V\cdot V\subset x^{-1}\cdot U$ , $B_{n}\subset V$ , and $d\in x\cdot B_{n}^{-1}$ . Then, $x\in d\cdot B_{n}\subset U$ .

Second, we suppose that for every covering of $G$ there is a countable subcovering. For each $n$ take a countable subcovering $\{c\cdot B_{n}|c\in C_{n}\}$ of the covering $\{x\cdot B_{n}|x\in G\}$ . The union over all $n$ forms a countable basis. Indeed, take again $x\in U$ . Choose $V$ and $B_{n}$ as before, and find $c\in C_{n}$ such that $x\in c\cdot B_{n}$ ; then $c\cdot B_{n}\subset U$ .

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Section 30: Problem 18 Solution

Section 30: Problem 18 Solution