Is there a topological group structure on the Baire Space?

Question

The sum of two irrationals might not be irrational so we can't use that; Also $\mathbb N$ is not a group so there's no obvious way to define a group structure in it seen as $\mathbb N^{\mathbb N}$ either. This answer gives some necessary conditions for a topological space to be possible to turn into a topological group. The Baire space clearly satisfies 1,4 and 5; intuitively i think it also satisfies 2 but i'm not sure how to prove it, and i have no idea for 3.

score 1 · Accepted Answer · answered Aug 05 '22 at 17:00

1

The Baire space is $\mathbb{Z}^{\mathbb{N}}$ with the product topology so it has an abelian topological group structure given by pointwise addition. $\mathbb{Z}$ can be replaced with any countable discrete group.

answered Aug 05 '22 at 17:00

Qiaochu Yuan

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That was embarassing haha – Carla only proves trivial prop Aug 05 '22 at 17:03
Since this question was trivial, is it okay if i edit it as i thought of a follow up question? – Carla only proves trivial prop Aug 05 '22 at 17:08
That's usually frowned upon but if it's not too different from the original question I guess that's fine. – Qiaochu Yuan Aug 05 '22 at 17:09

Is there a topological group structure on the Baire Space?

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