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In my work I encountered a map $f$ between two metric spaces $X$ and $Y$ that was not continuous (at least I couldn't prove it was), but I was able to prove that convergent sequences $(x_n)$ in $X$ were sent by $f$ to sequences lying in a compact set in $Y$ (in particular, any subsequence of $f(x_n)$ had a convergent subsequence).

Do these kind of maps have a name ?

Thanks.

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If nobody comes up with an extant name, I suggest "Bolzano maps". –  Felix Goldberg May 10 '12 at 9:10
    
You could be the Godfather, so you could name them anyway you please. –  Liviu Nicolaescu May 10 '12 at 9:31
    
Your condition on $f$ is equivalent to: $f(A)$ is pre-compact for any pre-compact $A$. So in the trend of "open map" and "closed map", I would call it a "pre-compact map". –  Ramiro de la Vega May 10 '12 at 16:23
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