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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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
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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. :-)