C*-Embedding Property and Stone-Cech Compactification

This is a continuation of an introduction of Stone-Cech compactification started in two previous posts (first post: A Beginning Look at Stone-Cech Compactification; second post: Two Characterizations of Stone-Cech Compactification). In this post, we present another characterization of the Stone-Cech compactification, that is, for any completely regular space X, X is C^*-embedded in its Stone-Cech compactification \beta X and that any compactification of X in which X is C^*-embedded must be \beta X. In other words, this property of C^*-embedding is unique to Stone-Cech compactification. We prove the following two theorems (U3 has two versions).

The links for other posts on Stone-Cech compactification can be found toward the end of this post.

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    Definition. Let Y be a space. Let A \subset Y. The subspace A is C^*-embedded in Y if every bounded continuous function f:A \rightarrow \mathbb{R} is extendable to a continuous \hat{f}:Y \rightarrow \mathbb{R}.

    Theorem C3
    Let X be a completely regular space. The space X is C^*-embedded in its Stone-Cech compactification \beta X.

    \text{ }

    Theorem U3.1
    Let X be a completely regular space. Let I=[0,1]. Let \alpha X be a compactification of X such that each continuous f:X \rightarrow I can be extended to a continuous \hat{f}:\alpha X \rightarrow I. Then \alpha X must be \beta X.

    \text{ }

    Theorem U3.2
    If \alpha X is any compactification of X that satisfies the property in Theorem C3 (i.e., X is C^*-embedded in \alpha X), then \alpha X must be \beta X.
    \text{ }

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Other Characterizations

Two other characterizations of \beta X are proved in the previous post (Two Characterizations of Stone-Cech Compactification).

    Theorem C1
    Let X be a completely regular space. Let f:X \rightarrow Y be a continuous function from X into a compact Hausdorff space Y. Then there is a continuous F: \beta X \rightarrow Y such that F \circ \beta=f.

    \text{ }

    Theorem U1
    If K is any compactification of X that satisfies condition in Theorem C1, then K must be equivalent to \beta X.
    \text{ }

    Theorem C2
    Let X be a completely regular space. Among all compactifications of the space X, the Stone-Cech compactification \beta X of the space X is the largest compactification.

    \text{ }

    Theorem U2
    The property in Theorem C2 is unique to \beta X. That is, if \alpha X is a compactification of X, then \alpha X must be equivalent to \beta X.

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Remark

The C theorems and the U theorems are a great tool to determine whether a given compactification is \beta X. Whenever a compactification \alpha X of a space X satisfies the property belonging to a C theorem, based on the corresponding U theorem, we know that this compactification \alpha X must be \beta X. For example, any compactification \alpha X that satisfies the function extension property in Theorem C1 must be \beta X. Th C^*-embedding property in Theorem C3 and Theorem U3 (both versions) is also a function extension property much like that in Theorems C1 and U1, but is easier to use. The reason being that we only need to extend a smaller class of continuous functions (i.e., to check whether functions from X into I=[0,1] can be extended), rather than checking all continuous functions from X to arbitrary compact spaces. As the following example below about \beta \omega_1 illustrates that the C^*-embedding in Theorem C3 and U3.1 can be used to describe \beta X explicitly.

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Proving Theorem U3.1 and Theorem U3.2

Let Y be a space. Let A be a subspace of X. Recall that A is C^*-embedded in Y if every bounded continuous function f:A \rightarrow \mathbb{R} can be extended to a continuous \hat{f}:Y \rightarrow \mathbb{R}.

Any bounded continuous function f: X \rightarrow \mathbb{R} can be regarded as f: X \rightarrow I_f where I_f is some closed and bounded interval. The C^*-embedding property in Theorem C3 is a function extension property like the one in Theorem C1, except that it deals with function from X into a specific type of compact spaces Y, namely the closed and bounded intervals in \mathbb{R}. Theorem C3 is a corollary of Theorem C1 (see below). So we only need to prove Theorem U3.1 and Theorem U3.2. Theorem U3.2 is a corollary of Theorem U3.1.

Proof of Theorem U3.1
By Theorem C2, we have \alpha X \le \beta X. So we only need to show \beta X \le \alpha X. To this end, we need to produce a continuous function H: \alpha X \rightarrow \beta X such that H \circ \alpha=\beta.

Let C(X,I) be the set of all continuous functions from X into I. For each f \in C(X,I), let I_f=I. Recall that \beta X is embedded in the cube \prod \limits_{f \in C(X,I)} I_f by the mapping \beta. For each f \in C(X,I), let \pi_f be the projection map from this cube into I_f.

Each f \in C(X,I) can be expressed as f=\pi_f \circ \beta. Thus by assumption, each f can be extended by \hat{f}: \alpha X \rightarrow I. Now define H: \alpha X \rightarrow \prod \limits_{f \in C(X,I)} I_f by the following:

    For each t \in \alpha X, H(t)=a=< a_f >_{f \in C(X,I)} such that a_f=\hat{f}(t)

For each x \in \alpha(X), we have H(\alpha(x))=\beta(x). Note that \hat{f} agrees with f on \alpha(X) since \hat{f} extends f. So we have H(\alpha(x))=a where a_f=\hat{f}(\alpha(x))=f(x) for each f \in C(X,I). On the other hand, by definition of \beta, we have \beta(x)=a where a_f=f(x) for each f \in C(X,I). Thus we have H \circ \alpha=\beta and the following:

    H(\alpha(X)) \subset \beta(X) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)

It is straightforward to verify that H is continuous. Note that \alpha(X) is dense in \alpha X. Since H is continuous, H(\alpha(X)) is dense in H(\alpha X). Thus we have:

    H(\alpha X)=\overline{H(\alpha(X))} \ \ \ \ \ \ \ \ \ \ \ \ \ (2)

Putting (1) and (2) together, we have the following:

    H(\alpha X)=\overline{H(\alpha(X))} \subset \overline{\beta(X)}=\beta X

Thus we can describe the map H as H: \alpha X \rightarrow \beta X. As noted before, we have H \circ \alpha=\beta. Thus \beta X \le \alpha X. \blacksquare

Proof of Theorem U3.2
Suppose \alpha X is a compactification of X such that X is C^*-embedded in \alpha X. Then every bounded continuous f:X \rightarrow I_f can be extended to \hat{f}:\alpha X \rightarrow I_f where I_f is some closed and bounded interval containing the range. In particular, this means every continuous f:X \rightarrow I can be extended. By Theorem U3.1, we have \alpha X \approx \beta X. \blacksquare

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Example

This is one example where we can use C^*-embedding to describe \beta X explicitly.

Let \omega_1 be the first uncountable ordinal. Let \omega_1+1 be the successor ordinal of \omega_1 (i.e. \omega_1 with one additional point at the end). Consider X=\omega_1 and Y=\omega_1+1 as topological spaces with the order topology derived from the well ordering of the ordinals. The space Y is a compactification of X. In fact Y is the one-point compactification of X.

It is well known that every continuous real-valued function on X is bounded (note that X here is countably compact and hence pseudocompact). Furthermore, every continuous real-valued function on X is eventually constant. This means that if f:X \rightarrow \mathbb{R} is continuous, for some \alpha < \omega_1, f is constant on the final segment X_\alpha=\left\{\rho < \omega_1: \rho>\alpha \right\} (see result B in The First Uncountable Ordinal). As a result, every continuous bounded real-valued function f:X \rightarrow \mathbb{R} can be extended to a continuous \hat{f}:Y \rightarrow \mathbb{R}. Then according to Theorem U3.2, \beta X=\beta \omega_1=Y=\beta \omega_1+1.

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Blog Posts on Stone-Cech Compactification

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Reference

  1. Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
  2. Willard, S., General Topology, Addison-Wesley Publishing Company, 1970.

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\copyright \ \ 2012

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