Lemma 5.25.1. Let f : X \to Y be a continuous map of topological spaces. Assume that f(X) is dense in Y and that Y is Hausdorff. Then the cardinality of Y is at most the cardinality of P(P(X)) where P is the power set operation.
5.25 Stone-Čech compactification
The Stone-Čech compactification of a topological space X is a map X \to \beta (X) from X to a Hausdorff quasi-compact space \beta (X) which is universal for such maps. We prove this exists by a standard argument using the following simple lemma.
Proof. Let S = f(X) \subset Y. Let \mathcal{D} be the set of all closed domains of Y, i.e., subsets D \subset Y which equal the closure of its interior. Note that the closure of an open subset of Y is a closed domain. For y \in Y consider the set
I_ y = \{ T \subset S \mid \text{ there exists } D \in \mathcal{D}\text{ with }T = S \cap D\text{ and }y \in D\} .
Since S is dense in Y for every closed domain D we see that S \cap D is dense in D. Hence, if D \cap S = D' \cap S for D, D' \in \mathcal{D}, then D = D'. Thus I_ y = I_{y'} implies that y = y' because the Hausdorff condition assures us that we can find a closed domain containing y but not y'. The result follows. \square
Let X be a topological space. By Lemma 5.25.1, there is a set I of isomorphism classes of continuous maps f : X \to Y which have dense image and where Y is Hausdorff and quasi-compact. For i \in I choose a representative f_ i : X \to Y_ i. Consider the map
\prod f_ i : X \longrightarrow \prod \nolimits _{i \in I} Y_ i
and denote \beta (X) the closure of the image. Since each Y_ i is Hausdorff, so is \beta (X). Since each Y_ i is quasi-compact, so is \beta (X) (use Theorem 5.14.4 and Lemma 5.12.3).
Let us show the canonical map X \to \beta (X) satisfies the universal property with respect to maps to Hausdorff, quasi-compact spaces. Namely, let f : X \to Y be such a morphism. Let Z \subset Y be the closure of f(X). Then X \to Z is isomorphic to one of the maps f_ i : X \to Y_ i, say f_{i_0} : X \to Y_{i_0}. Thus f factors as X \to \beta (X) \to \prod Y_ i \to Y_{i_0} \cong Z \to Y as desired.
Lemma 5.25.2. Let X be a Hausdorff, locally quasi-compact space. There exists a map X \to X^* which identifies X as an open subspace of a quasi-compact Hausdorff space X^* such that X^* \setminus X is a singleton (one point compactification). In particular, the map X \to \beta (X) identifies X with an open subspace of \beta (X).
Proof. Set X^* = X \amalg \{ \infty \} . We declare a subset V of X^* to be open if either V \subset X is open in X, or \infty \in V and U = V \cap X is an open of X such that X \setminus U is quasi-compact. We omit the verification that this defines a topology. It is clear that X \to X^* identifies X with an open subspace of X.
Since X is locally quasi-compact, every point x \in X has a quasi-compact neighbourhood x \in E \subset X. Then E is closed (Lemma 5.12.4 part (1)) and V = (X \setminus E) \amalg \{ \infty \} is an open neighbourhood of \infty disjoint from the interior of E. Thus X^* is Hausdorff.
Let X^* = \bigcup V_ i be an open covering. Then for some i, say i_0, we have \infty \in V_{i_0}. By construction Z = X^* \setminus V_{i_0} is quasi-compact. Hence the covering Z \subset \bigcup _{i \not= i_0} Z \cap V_ i has a finite refinement which implies that the given covering of X^* has a finite refinement. Thus X^* is quasi-compact.
The map X \to X^* factors as X \to \beta (X) \to X^* by the universal property of the Stone-Čech compactification. Let \varphi : \beta (X) \to X^* be this factorization. Then X \to \varphi ^{-1}(X) is a section to \varphi ^{-1}(X) \to X hence has closed image (Lemma 5.3.3). Since the image of X \to \beta (X) is dense we conclude that X = \varphi ^{-1}(X). \square
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