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As part of a casual self-study of topology, I have started fooling around with topological groups. I noticed that the article on Wikipedia mentioned that "weakening the continuity conditions" gives the definition for a semitopological group as one in which the functions $l_g,r_g:G\rightarrow G$ defined by $l_g(h)=gh$ and $r_g(h)=hg$ are continuous for each group element $g$.

Since this is supposedly a weaker condition, every topological group should be semitopological, but I don't understand how one would go about proving this. I feel like I should be able to use some sort of relationship between continuous functions from $G\times G$ to $G$ and continuous functions from $G$ to $G$, but I haven't been able to find anything of that nature.

So how can it be proven that topological groups are semitopological?

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2 Answers 2

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The restriction of a continuous function to a subspace is continuous. More generally, the composition of continuous functions is continuous (and restricting to a subspace is the special case that one of the continuous functions is the inclusion of a subspace).

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Basically, if the multiplication is continuous as a function of its two variables (as in a topological group) then the multiplication is continuous when you fix one of the variables and consider the resulting function on the other variable (as in a semitopological group). Informally, it is easier to be continuous with respect to just varying one of the two variables, hence every topological group is a semitopological group. For a proof, just notice that this is a case of restricting a function.

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