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I looked at quite a bit of questions on the site and didn't quite find this but apologies if it's here already.

I was wondering if any one knows if there exists an uncountable space and some Stone–Čech compactification that adds exactly two elements , i.e. $ \lvert \beta X \setminus X \lvert = 2$.

I thought about perhaps starting with $ \Omega_0 = [0, \omega_1) $ and sort of iterating compactifications if that makes sense. First adding $ \omega_1$ to the space which I believe gives us a one point compactification that is the same as the Stone–Čech compactification and then trying to do the same to the space that results from that ( i.e. $\Omega$). The problem is that if we apply the same one point compactification process I believe that since $ \Omega$ is already compact we will end up with a space which has a point isolated. Thus ruling out that the compactification is Stone–Čech .

Perhaps this space is not the one to try and work with. If anyone knows of an example that works or has any ideas that would be appreciated, thanks!

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    $\begingroup$ Retracting previous comment. You are correct that the one-point and Stone-Cech compactifications of $[0, \omega_1)$ are both equal to $[0, \omega_1]$. This is perhaps easiest to see via the characterization of the S-C compactification as a subspace of $C(X, [0,1])$, together with the fact that real-valued functions on $[0, \omega_1)$ are eventually constant. $\endgroup$ – Nate Eldredge Aug 6 '15 at 5:06
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    $\begingroup$ So with that in mind, can't we just take two copies of $[0,\omega_1)$? I.e., the space $X = [0, \omega_1) \times \{0,1\}$ with its product topology. It seems clear that the Stone-Cech ought to be $[0, \omega_1] \times \{0,1\}$. It cannot be a one-point compactification, for consider the function $f : X \to \{0,1\}$ defined by $f(\alpha, n) = n$, i.e. send one copy to 0 and the other to 1. Adding a single point at infinity can't give $f$ a continuous extension. Maybe a disappointingly boring example, but I think it works? $\endgroup$ – Nate Eldredge Aug 6 '15 at 5:13
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    $\begingroup$ @NateEldredge To see that the only (Hausdorff) compactification of $\omega_1$ is the one-point compactification, let $X$ be any compactification, and suppose there are two distinct points $a,b\in X\setminus\omega_1.$. Choose disjoint closed neighborhoods $A,B$ of $a,b$; then $A\cap\omega_1$ and $B\cap\omega_1$ are disjoint closed unbounded subsets of $\omega_1$ which is impossible. $\endgroup$ – bof Aug 6 '15 at 7:38
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Maybe a disappointingly boring example: take $X = [0, \omega_1) \times \{0,1\}$ with the product topology, i.e. the disjoint sum of two copies of $[0, \omega_1)$. Taking as given the fact that the Stone-Cech compactification (SCc) of $[0, \omega_1)$ is $[0, \omega_1]$, I claim the SCc of $X$ is $Y = [0, \omega_1] \times \{0,1\}$. Clearly $|Y \setminus X| =2$.

To check the details, we note that $Y$ is compact Hausdorff, and show that $Y$ has the appropriate universal property. So suppose $K$ is a compact Hausdorff space and $f : X \to K$ is continuous. We must show $f$ has a unique continuous extension $f' : Y \to K$. The uniqueness is clear because $X$ is dense in $Y$. For the existence, let $f_0, f_1 : [0, \omega_1) \to K$ be defined by $f_i(\alpha) = f(\alpha, i)$; these functions are continuous. So there are unique continuous extensions $f_i' : [0,\omega_1] \to K$. Set $f'(\alpha,i) = f_i'(\alpha)$. Then $f'$ is continuous by the pasting lemma, and for $(\alpha,i) \in X$ we have $f'(\alpha, i) = f_i'(\alpha) = f_i(\alpha) = f(\alpha_i)$, so $f'$ extends $f$.

Similarly, for any finite number $n$, one can take $X = [0, \omega_1) \times n$ and get a space with $|\beta X \setminus X| = n$. (As $[0,\omega_1] \times D$ is not compact for an infinite discrete space $D$, you can't replace $n$ with an infinite cardinal and expect the same thing to work.)

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  • $\begingroup$ Wow, very nice. Thank you Nate! $\endgroup$ – Jmaff Aug 6 '15 at 9:36
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There is an order topology which satisfies this. Simply put two copies of $[0,\omega_1)$ back to back (so the space would look like $(−\omega_1,\omega_1)$. Formally, let $X=2\times \omega_1$ and put the lexicographic ordering on X with the usual ordering reversed in the first factor. So $(i,x)<(j,y)$ in $X$ if $i=0$ and $j=1$, or if $i=j=0$ and $x>y$, or if $i=j=1$ and $x<y$. Endow $X$ with the order topology induced by this ordering. A continuous function on X is eventually constant towards both ends, and so it extends to $[-\omega_1,\omega_1]$, the space obtained by adding first element $\{-\omega_1\}$ and last element $\{\omega_1\}$ to $X$, which is a compactification of $X$. Thus $\beta X=[-\omega_1,\omega_1]$, and $|\beta X\setminus X|=2$.

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    $\begingroup$ This space is homeomorphic to $[0,\omega_1) \times 2$. $\endgroup$ – user642796 Aug 6 '15 at 7:46
  • $\begingroup$ Yes, it is just another way of looking at the space. $\endgroup$ – Forever Mozart Aug 6 '15 at 19:58

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