When is StoneČech compactification the same as onepoint 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), 419427:
...[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.
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Martin Sleziak
Updated on February 10, 2021Comments

Martin Sleziak over 2 years
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 StoneČech compactification and onepoint 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 StoneČech compactification $\beta X$ and onepoint compactification $\omega X$ are the same?

Martin Sleziak over 10 yearsThanks a lot! In case they are useful for someone, here are links to the Hewitt's paper: projecteuclid, doi:10.1215/S001270944701435X, MR, Zentralblatt. (I have looked in Engelking for places mentioning Alexandroff compactifications, so I missed this one.)

Henno Brandsma over 10 yearsThese spaces are called "almost compact" in some texts. I believe that this is also an exercise in Rings of continuous functions.

Martin Sleziak over 10 years@HennoBrandsma you're right: Exercise 6J in GillmanJerison is called almost compact spaces and it says that the following conditions are equivalent: (1) Of any two disjoint zerosets in $X$, at least one is compact. (2) $\beta XX\le1$. (3) $X\subset T$ implies $f(X)\subset f(T)$. (4) Every embedding of $X$ is a C${}^*$embedding. (5) Every embedding of $X$ is a Cembedding. (6) The only compactification of $X$ is $\beta X$. (7) Every embedding of any continuous image of $X$ is a Cembedding.

Martin over 10 years@MartinSleziak: You could add (8): The space $X$ admits a unique uniformity. See Chapter II, Exercise 11 (c) in J. R. Isbell, Uniform spaces.

Fernando Martin about 5 yearsIt's worth noting that the first uncountable ordinal is an example of a noncompact space satisfying the conditions above.