Stone-Cech Compactification of the Integers – Basic Facts

This is another post Stone-Cech compactification. The links for other posts on Stone-Cech compactification can be found below. In this post, we prove a few basic facts about \beta \omega, the Stone-Cech compactification of the discrete space of the non-negative integers, \omega=\left\{0,1,2,3,\cdots \right\}. We use several characterizations of Stone-Cech compactification to find out what \beta \omega is like. These characterizations are proved in the blog posts listed below. Let c denote the cardinality of the real line \mathbb{R}. We prove the following facts.

  1. The cardinality of \beta \omega is 2^c.
  2. The weight of \beta \omega is c.
  3. The space \beta \omega is zero-dimensional.
  4. Every infinite closed subset of \beta \omega contains a topological copy of \beta \omega.
  5. The space \beta \omega contains no non-trivial convergent sequence.
  6. No point of \beta \omega-\omega is an isolated point.
  7. The space \beta \omega fails to have many properties involving the existence of non-trivial convergent sequence. For example:
    \text{ }

    1. The space \beta \omega is not first countable at each point of the remainder \beta \omega-\omega.
    2. The space \beta \omega is not a Frechet space.
    3. The space \beta \omega is not a sequential space.
    4. The space \beta \omega is not sequentially compact.

    \text{ }

  8. No point of the remainder \beta \omega-\omega is a G_\delta-point.
  9. The remainder \beta \omega-\omega does not have the countable chain condition. In fact, it has a disjoint open collection of cardinality c.

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Characterization Theorems

For any completely regular space X, let C(X,I) be the set of all continuous functions from X into I=[0,1]. The Stone-Cech compactification \beta X is the subspace of the product space [0,1]^{C(X,I)} which is the closure of the image of X under the evaluation map \beta:X \rightarrow [0,1]^{C(X,I)} (for the details, see Embedding Completely Regular Spaces into a Cube).

The brief sketch of \beta \omega we present here is not based on the definition using the evaluation map. Instead we reply on some characterization theorems that are stated here (especially Theorem U3.1). These theorems uniquely describe the Stone-Cech compactification \beta X of a given completely regular space X. For example, \beta X satisfies the function extension property in Theorem C3 below. Furthermore any compactification \alpha X of X that satisfies the same property must be \beta X (Theorem U3.1). So a “C” theorem tells us a property possessed by \beta X. The corresponding “U” theorem tells us that there is only one compactification (up to equivalence) that has this property.

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    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 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.

    See Two Characterizations of Stone-Cech Compactification.
    _______________________________________________________________________________________
    \text{ }

    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.

    See C*-Embedding Property and Stone-Cech Compactification.
    _______________________________________________________________________________________
    \text{ }

The following discussion illustrates how we can use some of these characterizations theorem to obtain information about \beta X and \beta \omega in particular.

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Result 1 and Result 2

According to the previous post (Stone-Cech Compactification is Maximal), we have for any completely regular space X, \lvert \beta X \lvert \le 2^{2^{d(X)}} where d(X) is the density (the smallest cardinality of a dense set in X). With \omega being a countable space, \lvert \beta \omega \lvert \le 2^{2^{\omega}}=2^c.

Result 1 is established if we have 2^c \le \lvert \beta \omega \lvert. Consider the cube I^I where I is the unit interval I=[0,1]. Since the product space of c many separable space is separable (see Product of Separable Spaces), I^I is separable. Let S \subset I^I be a countable dense set. Let f:\omega \rightarrow S be a bijection. Clearly f is a continuous function from the discrete space \omega into I^I. By Theorem C1, f is extended by a continuous F:\beta \omega \rightarrow I^I. Note that the image F(\beta \omega) is dense in I^I since F(\beta \omega) contains the dense set S. On the other hand, F(\beta \omega) is compact. So F(\beta \omega)=I^I. Thus F is a surjection. The cardinality of I^I is 2^c. Thus we have 2^c \le \lvert \beta \omega \lvert.

From the same previous post (Stone-Cech Compactification is Maximal), it is shown that w(\beta X) \le 2^{d(X)}. Thus w(\beta \omega) \le 2^{\omega}=c. The same function F:\beta \omega \rightarrow I^I in the above paragraph shows that c \le w(\beta \omega) (see Lemma 2 in Stone-Cech Compactification is Maximal). Thus we have w(\beta \omega)=c \blacksquare

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Result 3

A space is said to be zero-dimensional whenever it has a base consisting of open and closed sets. The proof that \beta X is zero-dimensional comes after the following lemmas and theorems.

    Theorem 1
    Let X be a normal space. If H and K are disjoint closed subsets of X, then H and K have disjoint closures in \beta X.

Proof of Theorem 1
Let H and K be disjoint closed subsets of X. By the normality of X and by the Urysohn’s lemma, there is a continuous function g:X \rightarrow [0,1] such that g(H) \subset \left\{0 \right\} and g(K) \subset \left\{1 \right\}. By Theorem C3.1, g can be extended by G:\beta X \rightarrow [0,1]. Note that \overline{H} \subset G^{-1}(0) and \overline{K} \subset G^{-1}(1). Thus \overline{H} \cap \overline{K} = \varnothing. \blacksquare

    Theorem 2
    Let X be a completely regular space. Let H be a closed and open subset of X. Then \overline{H} (the closure of H in \beta X) is also a closed and open set in \beta X.

Proof of Theorem 2
Let H be a closed and open subset of X. Let K=X-H. Define \gamma:X \rightarrow [0,1] by letting \gamma(x)=0 for all x \in H and \gamma(x)=1 for all x \in K. Since both H and K are closed and open, the map \gamma is continuous. By Theorem C3, \gamma is extended by some continuous \Gamma:\beta X \rightarrow [0,1]. Note that \overline{H} \subset \Gamma^{-1}(0) and \overline{K} \subset \Gamma^{-1}(1). Thus H and K have disjoint closures in \beta X, i.e. \overline{H} \cap \overline{K} = \varnothing. Both H and K are closed and open in \beta X since \beta X=\overline{H} \cup \overline{K}. \blacksquare

    Lemma 3
    For every A \subset \omega, \overline{A} (the closure of A in \beta \omega) is both closed and open in \beta \omega.

Note that Lemma 3 is a corollary of Theorem 2.

    Lemma 4
    Let O \subset \beta \omega be a set that is both closed and open in \beta \omega. Then O=\overline{A} where A= O \cap \omega.

Proof of Lemma 4
Let A=O \cap \omega. Either O \subset \omega or O \cap (\beta \omega-\omega) \ne \varnothing. Thus A \ne \varnothing. We claim that O=\overline{A}. Since A \subset O, it follows that \overline{A} \subset \overline{O}=O. To show O \subset \overline{A}, pick x \in O. If x \in \omega, then x \in A. So focus on the case that x \notin \omega. It is clear that x \notin \overline{B} where B=\omega -A. But every open set containing x must contain some points of \omega. These points of \omega must be points of A. Thus we have x \in \overline{A}. \blacksquare

Proof of Result 3
Let \mathcal{A} be the set of all closed and open sets in \beta \omega. Let \mathcal{B}=\left\{\overline{A}: A \subset \omega \right\}. Lemma 3 shows that \mathcal{B} \subset \mathcal{A}. Lemma 4 shows that \mathcal{A} \subset \mathcal{B}. Thus \mathcal{A}= \mathcal{B}. We claim that \mathcal{B} is a base for \beta \omega. To this end, we show that for each open O \subset \beta \omega and for each x \in O, we can find \overline{A} \in \mathcal{B} with x \in \overline{A} \subset O. Let O be open and let x \in O. Since \beta \omega is a regular space, we can find open set V \subset \beta \omega with x \in V \subset \overline{V} \subset O. Let A=V \cap \omega.

We claim that x \in \overline{A}. Suppose x \notin \overline{A}. There exists open U \subset V such that x \in U and U misses \overline{A}. But U must meets some points of \omega, say, y \in U \cap \omega. Then y \in V \cap \omega=A, which is a contradiction. So we have x \in \overline{A}.

It is now clear that x \in \overline{A} \subset \overline{V} \subset O. Thus \beta \omega is zero-dimensional since \mathcal{B} is a base consisting of closed and open sets. \blacksquare

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Result 4 and Result 5

Result 5 is a corollary of Result 4. We first prove two lemmas before proving Result 4.

    Lemma 5
    For each infinite A \subset \omega, \overline{A} (the closure of A in \beta \omega) is a homeomorphic copy of \beta \omega and thus has cardinality 2^c.

Proof of Lemma 5
Let A \subset \omega. Let g:A \rightarrow [0,1] be any function (necessarily continuous). Let f:\omega \rightarrow [0,1] be defined by f(x)=g(x) for all x \in A and f(x)=0 for all x \in \omega-A. By Theorem C3, f can be extended by F:\beta \omega \rightarrow [0,1]. Let G=F \upharpoonright \overline{A}.

Note that the function G: \overline{A} \rightarrow [0,1] extends g:A \rightarrow [0,1]. Thus by Theorem U3.1, \overline{A} must be \beta A. Since A is a countably infinite discrete space, \beta A must be equivalent to \beta \omega. \blacksquare

    Lemma 6
    For each countably infinite A \subset \beta \omega-\omega such that A is relatively discrete, \overline{A} (the closure of A in \beta \omega) is a homeomorphic copy of \beta \omega and thus has cardinality 2^c.

Proof of Lemma 6
Let A=\left\{t_1,t_2,t_3,\cdots \right\} \subset \beta \omega -\omega such that A is discrete in the relative topology inherited from \beta \omega. There exist disjoint open sets G_1,G_2,G_3,\cdots (open in \beta \omega) such that for each j, t_j \in G_j. Since \beta \omega is zero-dimensional (Result 3), G_1,G_2,G_3,\cdots can be made closed and open.

Let f:A \rightarrow [0,1] be a continuous function. We show that f can be extended by F:\overline{A} \rightarrow [0,1]. Once this is shown, by Theorem U3.1, \overline{A} must be \beta A. Since A is a countable discrete space, \beta A must be equivalent to \beta \omega.

We first define w:\omega \rightarrow [0,1] by:

    \displaystyle w(n)=\left\{\begin{matrix}f(t_j)& \exists \ j \text{ such that } n \in \omega \cap G_j\\{0}&\text{otherwise} \end{matrix}\right.

The function w is well defined since each n \in \omega is in at most one G_j. By Theorem C3, the function w is extended by some continuous W:\beta \omega \rightarrow [0,1]. By Lemma 4, for each j, G_j=\overline{\omega \cap G_j}. Thus, for each j, t_j \in \overline{\omega \cap G_j}. Note that W is a constant function on the set \omega \cap G_j (mapping to the constant value of f(t_j)). Thus W(t_j)=f(t_j) for each j. So let F=W \upharpoonright \overline{A}. Thus F is the desired function that extends f. \blacksquare

Proof of Result 4
Let C \subset \beta \omega be an infinite closed set. Either C \cap \omega is infinite or C \cap (\beta \omega-\omega) is infinite. If C \cap \omega is infinite, then by Lemma 5, \overline{C \cap \omega} is a homeomorphic copy of \beta \omega. Now focus on the case that C_0=C \cap (\beta \omega-\omega) is infinite. We can choose inductively a countably infinite set A \subset C_0 such that A is relatively discrete. Then by Lemma 6 \overline{A} is a copy of \beta \omega that is a subset of C. \blacksquare

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Result 6

We prove that no point in the remainder \beta \omega-\omega is an isolated point. To see this, pick x \in \beta \omega-\omega and pick an arbitrary closed and open set O \subset \beta \omega with x \in O. Let V=O \cap (\beta \omega-\omega) (thus an arbitrary open set in the remainder containing x). By Lemma 4, O=\overline{A} where A=O \cap \omega. According to Lemma 5, O=\overline{A} is a copy of \beta \omega and thus has cardinality 2^c. The set V is O minus a subset of \omega. Thus V must contains 2^c many points. This means that \left\{ x \right\} can never be open in the remainder \beta \omega-\omega. In fact, we just prove that any open and closed subset of \beta \omega-\omega (thus any open subset) must have cardinality at least 2^c. \blacksquare

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Result 7

The results under Result 7 are corollary of Result 5 (there is no non-trivial convergent sequence in \beta \omega). To see Result 7.1, note that every point x in the remainder is not an isolated point and hence cannot have a countable local base (otherwise there would be a non-trivial convergent sequence converging to x).

A space Y is said to be a Frechet space if A \subset Y and for each x \in \overline{A}, there is a sequence \left\{ x_n \right\} of points of A such that x_n \rightarrow x. A set A \subset Y is said to be sequentially closed in Y if for any sequence \left\{ x_n \right\} of points of A, x_n \rightarrow x implies x \in A. A space Y is said to be a sequential space if A \subset Y is a closed set if and only if A is a sequentially closed set. If a space is Frechet, then it is sequential. It is clear that \beta \omega is not a sequential space.

A space is said to be sequentially compact if every sequence of points in this space has a convergent subsequence. Even though \beta \omega is compact, it cannot be sequentially compact.

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Result 8

Result 7.1 indicates that no point of remainder \beta \omega-\omega can have a countable local base. In fact, no point of the remainder can be a G_\delta-point (a point that is the intersection of countably many open sets). The remainder \beta \omega-\omega is a compact space (being a closed subset of \beta \omega). In a compact space, if a point is a G_\delta-point, then there is a countable local base at that point (see 3.1.F (a) on page 135 of [1] or 17F.7 on page 125 of [2]). \blacksquare

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Result 9

The space \beta \omega is a separable space since \omega is a dense set. Thus \beta \omega has the countable chain condition. However, the remainder \beta \omega-\omega does not have the countable chain condition. We show that there is a disjoint collection of c many open sets in \beta \omega-\omega.

There is a family \mathcal{A} of infinite subsets of \omega such that for every A,B \in \mathcal{A} with A \ne B, A \cap B is finite. Such a collection of sets is said to be an almost disjoint family. There is even an almost disjoint family of cardinality c (see A Space with G-delta Diagonal that is not Submetrizable). Let \mathcal{A} be such a almost disjoint family.

For each A \in \mathcal{A}, let U_A=\overline{A} and V_A=\overline{A} \cap (\beta \omega -\omega). By Lemma 3, each U_A is a closed and open set in \beta \omega. Thus each V_A is a closed and open set in the remainder \beta \omega-\omega. Note that \left\{V_A: A \in \mathcal{A} \right\} is a disjoint collection of open sets in \beta \omega-\omega. \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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