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A proper map is a continuous map such that compact sets have compact preimage. For locally compact Hausdorff spaces, it seems that a proper map can be extended to a continuous map between one-point compactifications. Is this extension unique?

If it is, we have following: a map $f:X\to Y$ is proper iff it has unique extension $f’:X’\to Y’$ (that sends infinity to infinity) is continuous. But I haven’t seen any reference or books using this kind of definition. Is this true?

Eric
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  • Extension to any compactification is unique if it exists. That's because by definition the domain embeds as dense subset of compactification. But it isn't true that existence of such extension implies being proper, e.g. any constant function extends to compactification, but generally these are not proper. – freakish Nov 13 '23 at 07:37
  • @freakish if $f\colon X\to Y$ is a constant function between locally compact noncompact spaces the extension $f'\colon X'\to Y'$ to one-point compactifications is not continuous, if $U\subseteq Y'$ is a nbhd of $\infty$ that avoids $f(X)$, then its preimage through $f'$ is ${\infty}$, which is not open. Indeed the result the question asks about is true – Alessandro Codenotti Nov 13 '23 at 09:29

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