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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?

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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.
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    $\begingroup$ Thanks a lot! In case they are useful for someone, here are links to the Hewitt's paper: projecteuclid, doi:10.1215/S0012-7094-47-01435-X, MR, Zentralblatt. (I have looked in Engelking for places mentioning Alexandroff compactifications, so I missed this one.) $\endgroup$ – Martin Sleziak Mar 21 '13 at 5:31
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    $\begingroup$ These spaces are called "almost compact" in some texts. I believe that this is also an exercise in Rings of continuous functions. $\endgroup$ – Henno Brandsma Mar 22 '13 at 19:27
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    $\begingroup$ @HennoBrandsma you're right: Exercise 6J in Gillman-Jerison is called almost compact spaces and it says that the following conditions are equivalent: (1) Of any two disjoint zero-sets in $X$, at least one is compact. (2) $|\beta X-X|\le1$. (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 C-embedding. (6) The only compactification of $X$ is $\beta X$. (7) Every embedding of any continuous image of $X$ is a C-embedding. $\endgroup$ – Martin Sleziak Mar 26 '13 at 9:27
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    $\begingroup$ @MartinSleziak: You could add (8): The space $X$ admits a unique uniformity. See Chapter II, Exercise 11 (c) in J. R. Isbell, Uniform spaces. $\endgroup$ – Martin Mar 27 '13 at 16:37
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    $\begingroup$ It's worth noting that the first uncountable ordinal is an example of a non-compact space satisfying the conditions above. $\endgroup$ – F M Sep 30 '18 at 16:22

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