Two Characterizations of Stone-Cech Compactification

This is the second post on Stone-Cech compactification (continuing from A Beginning Look at Stone-Cech Compactification). In this post, we establish two characterizations of Stone-Cech compactification. The first one is represented in the following diagram. The second one is that Stone-Cech compactification is maximal with respect to a certain partial order.

The first characterization is a central characteristic of Stone-Cech compactification. It is a function extension property that uniquely characterizes the Stone-Cech compactification of a completely regularly space. Here’s the diagram.

Figure 1

In this diagram, X is a completely regular space and \beta X is the Stone-Cech compactification of X where \beta is the homeomorphism mapping X onto \beta(X), which is dense in \beta X. The function f: X \rightarrow Y is an arbitrary continuous function where Y is compact. Then there exists a continuous function F:\beta X \rightarrow Y such that F restricted to \beta(X) is identical to the function f. In other words, if we think of X as a subset of \beta X, any continuous function from X to a compact space can be extended to all of \beta X. This function extension property is stated in Theorem C1 below.

    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. See Figure 1 above.
    \text{ }
    Theorem U1
    If K is any compactification of X that satisfies condition in Theorem C1, then K must be equivalent to \beta X.

Theorem C1 is the statement of the extension property described at the beginning. Theorem U1 states that this property is unique to \beta X. That is, of all the possible compactifications of X, only \beta X can satisfy Theorem C1.

For the other characterization, see Theorem C2 and Theorem U2 below.

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Defining Stone-Cech Compactification

The definition of \beta X=\beta_X X is given in this previous post (A Beginning Look at Stone-Cech Compactification) and is repeated here again for the sake of completeness. Let C(X,I) be the set of all continuous functions from X into I=[0,1]. For each g \in C(X,I), I_g=[0,1]. The map \beta_X:X \rightarrow \prod \limits_{g \in C(X,I)} I_g is defined by:

    For each x \in X, \beta_X(x)=t=< t_g >_{g \in C(X,I)} is the point t \in \prod \limits_{g \in C(X,I)} I_g such that t_g=g(x) for each g \in C(X,I) (i.e. the g^{th} coordinate of the point t is g(x)).

For the proof that \beta_X is a homeomorphism, see A Beginning Look at Stone-Cech Compactification. We have the following definition.

    Definition
    Under the map \beta_X, \beta_X(X) is the topological copy of X within the cube \prod \limits_{f \in C(X,I)} I_f. The Stone-Cech compactification of X is defined to be the closure of \beta_X(X) in the cube \prod \limits_{f \in C(X,I)} I_f, i.e., set \beta_X X=\overline{\beta_X(X)}.

When there is no ambiguity as to what the space X is, the embedding \beta_X is written as \beta and the compactification \beta_X X is written as \beta X (as in Figure 1 above). When more than one space is involved, we use subscripts to distinguish the embeddings, e.g., \beta_X and \beta_Y.

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Proof of Theorem U1

Let f:X \rightarrow Y be a continuous function from X into a compact Hausdorff space Y. Let \beta_X X be the Stone-Cech compactification of X where \beta_X is the homeomorphic embedding that defines \beta_X X. Since Y is a completely regular space, it has a Stone-Cech compactification \beta_Y Y, where \beta_Y is the homeomorphic embedding. We also define a map W from \prod \limits_{g \in C(X,I)} I_g into \prod \limits_{g \in C(Y,I)} I_k. We have the following diagram.

Figure 2

The desired function F will be defined by F=\beta_Y^{-1} \circ (W \upharpoonright \beta_X X). The rest of the proof is to define W and to show that this definition of F makes sense.

To define the function W, for each t \in \prod \limits_{g \in C(X,I)} I_g, let W(t)=a such that a_k=t_{k \circ f} (i.e. the k^{th} coordinate of W(t)=a is the (k \circ f)^{th} coordinate of t). With the definition of W, the diagram in Figure 2 commutes, i.e.,

    W \circ \beta_X=\beta_Y \circ f \ \ \ \ \ \ \ \ \ \ \ \ \ (1)

Starting with a point x \in X (the upper left corner of the diagram), we can reach the same point in the lower right corner regardless the path we take (W \circ \beta_X or \beta_Y \circ f). The following shows the derivation.

    One direction:
    x \in X

      \downarrow

    \beta_X(x)=t \text{ where } t_g=g(x) \ \forall \ g \in C(X,I)

      \downarrow

    W(t)=a \text{ where } a_k=t_{k \circ f}=(k \circ f)(x)=k(f(x)) \ \forall \ k \in C(Y,I)

    _________________________________
    The other direction:
    x \in X

      \downarrow

    f(x) \in Y

      \downarrow

    \beta_Y(f(x))=a \text{ where } a_k=k(f(x)) \ \forall \ k \in C(Y,I)

It is straightforward to verify that the map W is continuous. Based on (1) above, note that W(\beta_X(X)) \subset \beta_Y(Y). The following derivation shows that W(\beta_X X) \subset \beta_Y(Y).

    \displaystyle \begin{aligned} W(\beta_X X)&=W(\overline{\beta_X(X)}) \\&\subset \overline{W(\beta_X(X))} \ \ \ \ \text{ based on the continuity of } W\\&\subset \overline{\beta_Y(Y)} \ \ \ \ \ \ \ \ \ \ \text{ based on (1)}\\&=\beta_Y Y \\&=\beta_Y(Y) \ \ \ \ \ \ \ \ \ \ \text{ based on the compactness of Y} \end{aligned}

With the above derivation, we now know that the function W maps points of \beta_X X to points of \beta_Y(Y). So it makes sense to define F=\beta_Y^{-1} \circ (W \upharpoonright \beta_X X). Note that for each x \in X, we have:

    \displaystyle \begin{aligned} F(\beta_X(x))&=\beta_Y^{-1}(W(\beta_X(x)) \\&=\beta_Y^{-1}(\beta_Y(f(x))) \\&=f(x) \end{aligned}

Then we have F \circ \beta_X=f and F is the desired function. \blacksquare

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Compactifications

In order to prove Theorem U1, we first have a basic discussion on compactifications. Most importantly, we pin down what we mean when we say two compactifications of X are equivalent. In the process, we produce another characterization of Stone-Cech compactification (see Theorem C2 and Theorem U2 below).

Let X be a completely regular space. A pair (T,\alpha) is said to be a compactification of the space X if T is a compact Hausdorff space and \alpha:X \rightarrow T is a homeomorphism from X into T such that \alpha(X) is dense in T. More informally, a compactification of the space X can also be thought of as a compact space T containing a topological copy of the space X as a dense subspace.

Given a compactification (T,\alpha), we use the notation \alpha X rather than the pair (T,\alpha). By saying that \alpha X is a compactification of X, we mean \alpha X is the compact space T where \alpha is the homeomorphism embedding X onto \alpha(X).

The Stone-Cech compactification construction above is an example of a compactification. There can be more than one compactification of a given space X. For example, for X=\mathbb{R}, we have the Stone-Cech compactification \beta \mathbb{R}, which is a subspace of the cube \prod \limits_{f \in C(\mathbb{R},I)} I_f. The circle S^1=\left\{(x,y) \in \mathbb{R}^2: x^2+y^2=1 \right\} contains a copy of the real line \mathbb{R} as a dense subspace, as does the unit interval [0,1]. Thus both S^1 and I=[0,1] are also compactifications of \mathbb{R}. See A Beginning Look at Stone-Cech Compactification for a discussion of these examples.

We say that compactifications \alpha_1 X and \alpha_2 X are equivalent (we write \alpha_1 X \approx \alpha_2 X) if there exists a homeomorphism f: \alpha_1 X \rightarrow \alpha_2 X such that f \circ \alpha_1= \alpha_2. In other words, the following diagram commutes.

Figure 3

Essentially, two compactifications \alpha_1 X and \alpha_2 X of X are equivalent if there is a homeomorphism f between the two and if each x \in X is mapped by f to itself, i.e., \alpha_1(x) is mapped to \alpha_2(x).

For a given completely regular space X, let \mathcal{C}(X) be the class of all compactifications of X. We define a partial order \le on \mathcal{C}(X). For \alpha_1 X and \alpha_2 X, both in \mathcal{C}(X), we say that \alpha_2 X \le \alpha_1 X if there is a continuous function f:\alpha_1 X \rightarrow \alpha_2 X such that f \circ \alpha_1=\alpha_2. See Figure 4 below.

Figure 4

The following theorem ties the partial order \le to the equivalence relation \approx for compactifications.

    Theorem 1
    Let \alpha_1 X and \alpha_2 X be two compactifications of X. Then \alpha_1 X \le \alpha_2 X and \alpha_2 X \le \alpha_1 X if and only if \alpha_1 X \approx \alpha_2 X.

Proof of Theorem 1
\Rightarrow With \alpha_2 X \le \alpha_1 X, there exists continuous f_1:\alpha_1 X \rightarrow \alpha_2 X such that

    f_1 \circ \alpha_1=\alpha_2 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (A1)

With \alpha_1 X \le \alpha_2 X, there exists continuous f_2:\alpha_2 X \rightarrow \alpha_1 X such that

    f_2 \circ \alpha_2=\alpha_1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (A2)

Applying f_2 to (A1), we have f_2 \circ f_1 \circ \alpha_1=f_2 \circ \alpha_2. Applying (A2) to this result, we have

    f_2 \circ f_1 \circ \alpha_1=\alpha_1 \ \ \ \ \ \ \ \ \ \ \ (A3)

Note that f_2 \circ f_1 is a map from \alpha_1 X into \alpha_1 X. The equation (A3) indicates that when f_2 \circ f_1 is restricted to \alpha_1(X), it is the identity map. Thus f_2 \circ f_1 agrees with the identity map on the dense set \alpha_1(X). This implies that \alpha_1(X) must agree with the identity map on all of \alpha_1 X.

Likewise we can see that f_1 \circ f_2 must equal to the identity map on \alpha_2 X. So f_1:\alpha_1 X \rightarrow \alpha_2 X is a homeomorphism and it follows that \alpha_1 X and \alpha_2 X are equivalent compactifications of X.

\Leftarrow This direction is straightforward. Let f:\alpha_1 X \rightarrow \alpha_2 X a homeomorphism that makes \alpha_1 X and \alpha_2 X equivalent (as described by Figure 3). Then the map f implies \alpha_2 X \le \alpha_1 X and the map f^{-1} implies \alpha_1 X \le \alpha_2 X. \blacksquare

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Another Characterization of the Stone-Cech Compactification

The next theorem says that the Stone-Cech compactification is the maximal compactification with respect to the partial order \le defined here. Furthermore, this property is unique (there is only one maximal compactification up to equivalence). This result will simplify the work when we need to show that a given compactification is equivalent to \beta X.

    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 maximal with respect to the partial order \le.

    \text{ }

    Theorem U2
    The property in Theorem C2 is unique to \beta X. That is, if, among all compactifications of the space X, \alpha X is maximal with respect to the partial order \le, then \alpha X \approx \beta X.

Proof Theorem C2
Let \alpha X be any compactification of X. Consider the continuous map \alpha:X \rightarrow \alpha X. By Theorem C1, \alpha can be extended to \beta X. In other words, there exists a continuous F: \beta X \rightarrow \alpha X such that F \circ \beta = \alpha. The existence of the map F implies that \alpha X \le \beta X. \blacksquare

Proof Theorem U2
Let \alpha X be another maximal compactification of X. This implies that \beta X \le \alpha X. By Theorem C2, we have \alpha X \le \beta X. By Theorem 1, \alpha X must be equivalent to \beta X. \blacksquare

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Proof of Theorem U1

We are now ready to prove Theorem U1.

Proof of Theorem U1
Let \alpha X be a compactification of X that satisfies the extension property in Theorem C1. In light of Theorem C2, we have \alpha X \le \beta X. So we only need to show \beta X \le \alpha X. Consider the map \beta: X \rightarrow \beta X. By the assumption that \alpha X satisfies the extension property in Theorem C1, there exists a continuous function F:\alpha X \rightarrow \beta X such that F \circ \alpha=\beta. The existence of F implies that \beta X \le \alpha X. By Theorem 1, \alpha X must be equivalent to \beta X. \blacksquare

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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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