[Math] When is Stone-Čech compactification the same as one-point compactification

alexandroff-compactificationcompactificationcompactnessgeneral-topology

For the space $\omega_1$ (with the order topology) we have $\beta\omega_1=\omega_1+1$ (or $\beta[0,\omega_1)=[0,\omega_1]$, if you prefer this notation), i.e., it is an example of a space for which
Stone-Čech compactification
and one-point compactification
(a.k.a. Alexandroff compactification) coincide. (See, for example, this answer and this blog.)

Is there some known characterization of topological spaces such that Stone-Čech compactification $\beta X$ and one-point compactification $\omega X$ are the same?

Best Answer

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

  1. The space $X$ has a unique (up to equivalence) compactification.
  2. The space $X$ is compact or $| \beta X \setminus X | = 1$.
  3. If two closed subsets of $X$ are completely separated, then at least one of them is compact.
Related Question