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?
Properties of Topological Groups
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general-topology
topological-groups
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0You mean if the topology is Hausdorff? – 2012-06-23
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0What 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. – 2012-06-24
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5I 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. – 2012-06-24
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0@Keenan: I edited the text "group over a topology" to express what has to be the intended meaning. – 2012-06-24
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0@tomasz: could you please explain why that is true? – 2012-06-24
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0The way it is stated in the question, I don't even think that it is true, and I certainly can't prove that it is, so I deleted my answer. – 2012-06-30