1

Is it true that a countable LCA group can only be discrete ?

This question is related to a comment here :

https://math.stackexchange.com/questions/1168044/a-theorem-on-lca-group-is-the-uncountability-necessary?noredirect=1#comment2380628_1168044

Community
  • 1
M.G
  • 3,799

1 Answers1

2

It appears that this is correct: the only countable LCA groups are discrete. Using the first structure theorem for LCA groups, we can reduce this to the case where the group $G$ is compact. According to this, there is a theorem:

"Every countable compact Hausdorff space is homeomorphic to some well-ordered set with the order topology."

However, the only well-ordered sets $X$ which can support a topological group structure are subsets of $\mathbb N$ (and hence are discrete): otherwise, some points of $X$ will be limit points and others will not, contradicting that $X$ should act transitively (by homeomorphisms) on itself by multiplication.

Community
  • 1
Brent Kerby
  • 5,709