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A quasitopological group is a group $G$ with topology such that multiplication $G\times G\rightarrow G$ is continuous in each variable (i.e. all translations are continuous) and inversion $G\rightarrow G$ is continuous. Sometimes these are called semitopological or semicontinuous groups. What (if it exists) is an example of a quasitopological group such that at least one of the $n$-th power maps $g\mapsto g^{n}$ (for $n\geq 2$) is discontinuous?

I am pretty sure such an example exists but I am having a hard time finding one in the literature.

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Maybe, the following topology on the plane works: a base at 0 is formed by the usual neighborhoods at 0 in the plane minus a convenient subset of the diagonal, e.g. the sequence 1/3^n (and -1/3^n).

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Thanks, I think this works! It seems you could do the same thing in $\mathbb{R}$ too (or just consider the diagonal subgroup of what you defined with the subspace topology). –  Jeremy Brazas Jul 12 '10 at 13:59

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