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Does anyone know what the problem is?
A:
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Q:
Is there a better way of showing that the Halmos topological group has a countable subbase than the approach described here?
Here is the problem.
Let $G$ be a Halmos topological group. Show that $G$ has a countable subbase.
My approach:
Let $\mathcal B=\{B(x,r): x\in G, r\in \mathbb Q^+\}$. It’s clear that $\mathcal B$ is countable. We need to show $\mathcal B$ is a subbase. The proof is a tedious exercise I will skip to save time.
A:
The problem is just a special case of a general theorem of topology and group theory which says that if $G$ is a Hausdorff second countable (i.e., compact, connected, locally compact) locally connected group, then $G$ is locally compact Hausdorff (i.e., it has a countable base, or equivalently, it has a pre-compact neighborhood base). If one wants to have a „quick and dirty” proof of this fact, one can simply show that since Hausdorff compact second countable locally connected groups are metrizable, they are first countable (see, e.g., Billingsley, Topological groups, pp. 220-221).
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