Two basic questions about topological group theory

Question

For a topological group, I'd like to know whether

1.there exist a topological group G which is a Hausdorff space but does not satisfies the first countable axiom or

2.there exist a topological group G which is not a Hausdorff space and does not satisfies the first countable axiom.

I really find it difficult for me. Help me please. I can't work it out so far.

Please give me two examples about them. Thank you very much.

Answer 1

1. The product of uncountably many discrete groups of order two is a compact Hausdorff topological group which is not first countable.

2. To get a topological group which is neither Hausdorff nor first countable, take the product of any non-first-countable topological group (such as the one in the previous example) with any non-Hausdorff topological group (such as a group of order two with the "indiscrete" topology).