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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.

David Chan
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1 Answers1

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  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).

bof
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  • Thank you very much. I still have a little/bit of confused about "1.The product of uncountably many discrete groups of order two is a compact Hausdorff topological group which is not first countable." Why it's not the first countable? Please show me about it. – David Chan Jan 27 '14 at 01:37
  • 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 05:01
  • Ah ha, I have known about that. Thank you very much. – David Chan Jan 28 '14 at 07:24
  • If you have received a satisfactory answer to your question, you may wish to consider accepting it. – bof Jan 28 '14 at 08:21