TITLE: Which spaces can be seen as to be the one-point compactification of some other space
QUESTION [5 upvotes]: Given a topological space $X$, under what conditions is $X$ the one-point compactification of some other space $Y$? Obviously $X$ must be compact but it seems unlikely to me that every compact space arises from one-point compactification.

REPLY [5 votes]: Any compact Hausdorff space $X$ with a non-isolated point $p\in X$ (so $\{p\}$ is not open) is homeomorphic to the one-point compactification of $Y:=X\setminus \{p\}$ by standard theorems. 
A finite discrete space shows why we need the condition for a non-isolated point.