Two basic questions about topological group theory

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.

• It's not first countable because, if you form the intersection of countably many basic neighborhoods of $0$, you don't get $\{0\}$. Alternatively, if you take the product of $2^{\aleph_0}$ copies of $\{0,1\}$, you get a separable Hausdorff space with $\gt2^{\aleph_0}$ points; it can't be first countable, because a first countable separable Hausdorff space has $\le2^{\aleph_0}$ points. – bof Jan 27 '14 at 5:01