## Topological groups (Direct Product and Product Spaces)

Up until now we’ve discussed how the subspace topology reacts with subgroups and how the quotient topology interacts with the quotient group, it seems like a natural progression to then discuss how the product topology reacts with the direct product. So, as in the past we firstly need to verify that the two do agree, namely:

**Theorem:** Let be topological groups. Then, is a topological group with the direct product group structure and product topology.

**Proof: **Clearly is a set with both a group theoretic and topological structure and so it remains to show that the map

given by

is continuous. But, note that

And that

given by

being the product of continuous maps is continuous. The conclusion immediately follows.

There are canonical topological epimorphisms from into . Namely:

**Theorem: **Let

Then, is an open topological epimorphism.

**Proof:** The fact that it’s open, surjective, and continuous follows since topologically is merely the canonical projection of a product space onto it’s th coordinate. Thus, it remains to show that it’s a homomorphism. But, this is a routine calculation.

**Theorem:** Let be as above. Then,

from where it follows from a previous theorem that

.

Now, so we can use it we should mention in passing that:

**Theorem: ** where is any bijeciton.

**Proof: **Obvious.

In fact, that is all I wanted to mention about them. Unfortunately the interesting stuff about the direct product of topological groups that is interesting isn’t “accessible” and would take too much time to really discuss. Thus, for once I have a post which is less than one-thousand words. Enjoy.

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