A Beginning Look at Stone-Cech Compactification

For every completely regular space X, there exists a unique compact space \beta X containing X such that (1) \beta X has a dense subspace that is a topological copy of X and that (2) if X is considered as a subspace of \beta X, any continuous function f: X \rightarrow Y from X into a compact space Y can be extended to all of \beta X. The compact space \beta X is said to be the Stone-Cech compactification of X. We indicate how \beta X is constructed. The construction is done by embedding a completely regular space into a cube. To provide a glimpse of what \beta X might look like, we also take a brief look at \beta \mathbb{R}.

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

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Construction

According to a theorem by Tychonoff from 1930, every completely regular space can be embedded in a cube. Embedding into a cube is one way Stone-Cech compactification is constructed. Let X be a completely regular space. Let I=[0,1] be the unit interval in the real line \mathbb{R}. Let C(X,I) be the set of all continuous real-valued functions defined on the space X. The cube for which X is embedded is I^{\mathcal{K}}, the product of \mathcal{K} copies of I where \mathcal{K} is the cardinality of C(X,I). We can also view I^{\mathcal{K}} as the product space \prod \limits_{f \in C(X,I)} I_f where each I_f=I. We can represent each point in the cube as a function t:C(X,I) \rightarrow I or as a sequence < t_f >_{f \in C(X,I)} such that each term (or coordinate) t_f \in I=[0,1].

The embedding from X into the cube \prod \limits_{f \in C(X,I)} I_f is the evaluation map \beta_X which is defined as: for each x \in X, \beta_X(x)=t=< t_f >_{f \in C(X,I)} is the point in the cube such that for each f \in C(X,I), t_f=f(x). This map \beta_X is shown to be a homeomorphism from X into \prod \limits_{f \in C(X,I)} I_f. What makes \beta_X a homeomorphism stems from the fact that in a completely regular space, there are enough bounded real-valued continuous functions to separate points from closed sets (see Embedding Completely Regular Spaces into a Cube). 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)}.

According to the Tychonoff theorem, the cube \prod \limits_{f \in C(X,I)} I_f, being a product of compact spaces, is a compact space. Thus \beta_X X, being a closed subspace of the cube, is a compact space. Furthermore, \beta_X X contains a topological copy of X as a dense subspace.

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.

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Characterization

Let X be a completely regular space. A compactification of X is a compact space K that has a dense subspace that is a topological copy of X. For a given X, there can be many compactifications of X. The Stone-Cech compactification \beta X has characteristics that are not shared by other compactifications of X. One such characteristic is stated at the beginning of the post and is repeated in the following two theorems.

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

Figure 1

When the continuous function F in Theorem C1 is restricted to to \beta(X), it is identical to the function f. If we think of X as a subset of \beta X, F extends f. Thus Theorem U1 essentially says that any continuous function from X into any compact space can be extended to all of \beta X. Theorem U1 says that the property stated in Theorem C1 uniquely characterizes \beta X. This means that any compactification of X that satisfies Theorem C1 must be equivalent to \beta X.

The proofs of Theorem C1 and Theorem U1 are found in the next post (Two Characterizations of Stone-Cech Compactification).

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Examples

We now look at \beta \mathbb{R}. We rely on Theorem C1 to provide some insight about \beta \mathbb{R}.

Note that S^1 and [0,1] are also compactifications of \mathbb{R}, where S^1 is the unit circle S^1=\left\{(x,y) \in \mathbb{R}^2: x^2+y^2=1 \right\}. First, we show the extension property described in Theorem C1 does not hold for both S^1 and [0,1]. Thus these two compactifications of \mathbb{R} cannot be \beta \mathbb{R}.

Consider the function \Phi(x):\mathbb{R} \rightarrow [0,1] defined by:

    \displaystyle \Phi(x)=\int_{-\infty}^x \ \frac{1}{\sqrt{2 \pi}} \ e^{\frac{- t^2}{2}} \ dt

Note that \Phi(x) is the cumulative distribution function (CDF) of the standard normal distribution. The following figure is the graph.

Figure 2

Note that S^1 is the one-point compactification of \mathbb{R}. Since \lim \limits_{x \rightarrow +\infty} \Phi(x)=1 and \lim \limits_{x \rightarrow -\infty} \Phi(x)=0, we cannot extend the function \Phi to a continuous function defined on S^1. Thus, S^1 as a compactification of \mathbb{R} cannot satisfy Theorem C1. It is clear that \beta \mathbb{R} cannot be S^1.

The closed unit interval [0,1] is the two-point compactification of \mathbb{R}. Now consider the function s(x)=sin(\frac{1}{x}) defined on the open interval (0,1), which is a topological copy of \mathbb{R}. It is impossible to extend s(x) to a continuous function defined on [0,1]. So to construct \beta \mathbb{R}, it take more than adding two additional points. In fact, the following discussion shows that \beta \mathbb{R} has uncountably many points in the remainder \beta \mathbb{R}-\mathbb{R}.

Consider the function the sine function w(x)=\text{sin}(x), which maps \mathbb{R} onto [-1,1]. Based on Theorem C1, w(x) can be extended to a function G: \beta \mathbb{R} \rightarrow [-1,1].

For each t \in [-1,1], let A_t=\left\{x \in \mathbb{R}: w(x)=\text{sin}(x)=t \right\}. For example, A_0 is the set \displaystyle \left\{n \pi: n=0, \pm 1, \pm 2, \pm 3, \cdots \right\}. Furthermore, for each t \in [-1,1], let B_t=G^{-1}(t), which is a compact set in \beta \mathbb{R}.

Consider \mathbb{R} as a subset of \beta \mathbb{R}. We have A_t \subset B_t for each t \in [-1,1]. Each A_t is a discrete set in \mathbb{R}. However, each A_t is not discrete in \beta \mathbb{R} (A_t is an infinite subset of a compact set in \beta \mathbb{R}). Thus B_t-A_t \ne \varnothing for each t \in [-1,1].

If we think of constructing a compactification as the process of adding points to X to form a compact space, the non-empty sets B_t-A_t show that at minimum we are adding continuum many points to form \beta \mathbb{R} (adding as many points as there are points in [-1,1] but in reality more than continuum many points are added). Note that for t \ne p, B_t and B_p are disjoint (they are separated by disjoint open sets in \beta \mathbb{R} through inverse images of the function G).

In fact, the cardinality of \beta \mathbb{R} is larger than continuum. Specifically, \lvert \beta \mathbb{R} \lvert=2^c, where c is the cardinality of \mathbb{R} and 2^c is the cardinality of the set of all subsets of \mathbb{R}. For this fact, see Corollary 3.6.12 in [1] or Exercise 19.H in [2].

The large size of \beta \mathbb{R} also tells us something about it topologically. It is a well known theorem in general topology that the cardinality of every first countable compact Hausdorff space is at most continuum (see Corollary 3.1.30 in [1] or see The cardinality of compact first countable spaces, I). Thus \beta \mathbb{R} cannot be first countable. The points at which it is not first countable are the points in the remainder \beta \mathbb{R}-\mathbb{R}. Not being a first countable, \beta \mathbb{R} is not metrizable.

Much more can be said about \beta \mathbb{R} and other Stone-Cech compactification \beta X. This brief walk in \beta \mathbb{R} shows that the Stone-Cech compactification can be quite large and quite different topologically even when the starting space is the familiar Euclidean space. In subsequent discussions, we will see that nice properties such as first countability, second countability and metrizability do not carry over from X to \beta X.

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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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