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?


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.
  • 3
    $\begingroup$ 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
  • 3
    $\begingroup$ 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
  • 3
    $\begingroup$ @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
  • 3
    $\begingroup$ @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
  • 2
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.