Show that every continuous real-valued function defined on \(S_\Omega\) is “eventually constant.” [Hint: First prove that for each \(\epsilon\), there is an element \(\alpha\) of \(S_\Omega\) such that \(\abs{f(\beta) - f(\alpha)} < \epsilon\) for all \(\beta > \alpha\). Then let \(\epsilon = 1/n\) for \(n \in \mathbb{Z}_+\) and consider the corresponding points \(\alpha_n\).]
Show that the one-point compactification of \(S_\Omega\) and the Stone-Čech compactification are equivalent.
Conclude that every compactification of \(S_\Omega\) is equivalent to the one-point compactification.
Solution
Write \(S_\Omega=[0,\Omega)\), where \(\Omega\) is the first uncountable ordinal.
- We first show that for every \(\varepsilon>0\) there is \(\alpha<\Omega\) such that \[ |f(\beta)-f(\gamma)|<\varepsilon \qquad(\beta,\gamma>\alpha). \] If not, recursively choose ordinals \[ \alpha_1<\beta_1<\alpha_2<\beta_2<\cdots<\Omega \] with \[ |f(\alpha_n)-f(\beta_n)|\ge\varepsilon. \] The supremum \[ \lambda=\sup_n\beta_n \] is still countable, hence \(\lambda<\Omega\). Both sequences \((\alpha_n)\) and \((\beta_n)\) converge to \(\lambda\) in the order topology, so continuity of \(f\) gives \[ f(\alpha_n)\to f(\lambda),\qquad f(\beta_n)\to f(\lambda), \] contradicting the displayed lower bound.
For each \(n\ge1\), choose \(\alpha_n\) so that the oscillation of \(f\) on \((\alpha_n,\Omega)\) is \(<1/n\). Let \[ \alpha=\sup_n\alpha_n<\Omega. \] If \(\beta,\gamma>\alpha\), then \[ |f(\beta)-f(\gamma)|<1/n \] for every \(n\), hence \(f(\beta)=f(\gamma)\). Thus \(f\) is eventually constant.
Let \(S_\Omega^*=S_\Omega\cup\{\Omega\}\) be the one-point compactification, which is the ordinal interval \([0,\Omega]\). Every bounded continuous real-valued function on \(S_\Omega\) is eventually constant by (a), so it extends continuously to \(\Omega\) by assigning that eventual constant value there. By the Stone–Čech characterization via extension of bounded continuous real-valued functions, \(S_\Omega^*\) is equivalent to \(\beta S_\Omega\).
Let \(Y\) be any compactification of \(S_\Omega\). By maximality of the Stone–Čech compactification there is a continuous surjection \[ q:\beta S_\Omega\longrightarrow Y \] that is the identity on \(S_\Omega\). By (b), \(\beta S_\Omega-S_\Omega\) consists of the single point \(\Omega\). Since \(q\) fixes every point of \(S_\Omega\), surjectivity implies that \(Y-S_\Omega\) has at most one point; it is nonempty because \(S_\Omega\) is not compact. Thus \(Y\) is a one-point compactification. One-point compactifications of locally compact Hausdorff spaces are unique up to equivalence, so every compactification of \(S_\Omega\) is equivalent to \(S_\Omega^*\).