Is it true that the one-point-compactification of a Polish space is again a Polish space? I am currently learning something about Feller Processes and I think at some point this is needed.
2026-04-06 12:38:19.1775479099
one-point-compactification of a Polish space is a Polish space?
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Yes, this is true: if a Polish space $X$ is locally compact then $\alpha X$ will be compact Hausdorff and as $X$ is separable (and thus second countable), the same holds for $\alpha X$, and a second countable compact Hausdorff space is Polish.
For a non-locally compact Polish space, $\alpha X$ will not even be Hausdorff, let alone Polish.