When is Stone-Čech compactification the same as one-point compactification?
The following is from a problem in Engelking (Problem 3.12.16, p.234), and it credited to E. Hewitt, Certain generalizations of the Weierstrass approximation theorem, Duke Math. J. 14 (1947), 419-427:
...[F]or every Tychonoff space $X$ the following conditions are equivalent
- The space $X$ has a unique (up to equivalence) compactification.
- The space $X$ is compact or $| \beta X \setminus X | = 1$.
- If two closed subsets of $X$ are completely separated, then at least one of them is compact.