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I'm working though William Basener's Topology and Its Applications and I have come across a problem I can't solve. The book defines a topological group as a group equipped with a topology where for each element $a$, $L_{a} (x) = a + x$ and $R{a}(x) = x + a$ are both continuous. I need to prove that if the topology underlying the group is Hausdorff then $f(x, y) = x - y$ is continuous iff those all such functions $L$ and $R$ are continuous. Any ideas?

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You mean if the topology is Hausdorff? – Ehsan M. Kermani Jun 23 '12 at 23:43
What do you mean by "a group over a topology?" Also, first you say that continuity of the functions $La$ and $Ra$ is part of your definition, but in your question, you mention a Hausdorff group, so presumably a topological group, and ask about the continuity of those functions. This doesn't quite make sense. – Keenan Kidwell Jun 24 '12 at 0:33
I think you meant to ask about proof that a group with Hausdorff topology is a topological group iff $f(x,y)=xy^{-1}$ is continuous. For that you need not only separate contunuity of multiplication, but also joint continuity and continuity of the inverse. This follows from the fact that the diagonal is closed for Hausdorff spaces, iirc. – tomasz Jun 24 '12 at 1:08
@Keenan: I edited the text "group over a topology" to express what has to be the intended meaning. – KCd Jun 24 '12 at 3:46
@tomasz: could you please explain why that is true? – Parakee Jun 24 '12 at 14:23

The definition of topological group which you wrote is highly non-standard (I never saw it before) and the claim which you tried to prove is false. The situation is described at the first page of my paper. :-)

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