𝔉𝔲𝔯𝔪𝔞𝔫 𝔘𝔫𝔦𝔳𝔢𝔯𝔰𝔦𝔱𝔶 Electronic Journal of Undergraduate Mathematics Volume 2, 1996 1-5

COMPACTIFICATIONS OF TOPOLOGICAL SPACES

JAY BLANKESPOOR AND JOHN KRUEGER

Abstract. Given a locally compact, Hausdorff space, X, there are several ways to compactify it. We examine two of these compactifications – the onepoint compactification, X̂, and the Stone-Čech compactification, β(X) – and give conditions which guarantee that X̂ ≠ β(X). We also give new examples of topological spaces for which X̂ = β(X).

1. Introduction

Every calculus student knows that a continuous, real-valued function of a real variable is guaranteed of attaining a maximum or a minimum only on a closed and bounded set. Students of topology and analysis learn that continuous functions behave very nicely on compact sets. But when first presented with the formal definition of compact set, it is unlikely that they have much appreciation for the fact that the definition evolved as the answer to the question: “What is it about closed and bounded subsets of the real line that makes continuous functions defined on them behave so well?” It is important to recognize that it is precisely this “open cover” condition that forces the “nice” behavior of continuous functions defined on such spaces.

If a topological space is not compact, natural questions to ask are: “How ‘close’ is it to being compact?,” or “Can we ‘add’ anything to it to make it compact?” If the space is locally compact and Hausdorff, the answer to the second question is “yes”. (See, e.g., ([1] p. 183)). Generally, in fact, there are several different ways to compactify a locally compact Hausdorff space. In this paper we will investigate the “smallest” (one-point) compactification of X and the “largest” (Stone-Čech) compactification of X. We will give examples of spaces for which these two compactifications are the same (and thus X has only one compactification) and give conditions which guarantee that these two compactifications are different.

2. Compactification of a space, X

Formally, we say that a compact Hausdorff space Y is a compactification of X if X is a subspace of Y such that X is dense in Y. Two compactifications Y_{1} and Y_{2} of X are called equivalent if there is a homeomorphism h : Y_{1} → Y_{2} such that h(x) = x for all x ∈ X. Does every locally compact Hausdorff space have a compactification? We answer this question affirmatively in the following sections.

Received by the editors January, 1996.

1991 Mathematics Subject Classification. 54m.

Key words and phrases. Locally compact, Hausdorff space; Lindelof space; One point compactification; Stone-Čech compactification; Tychonoff theoreom.

.

3. The one-point compactification of X

The “easiest” or “smallest” compactification of a locally compact Hausdorff space is the one-point compactification which we denote by X̂. X̂ is constructed as follows: let the symbol ∞ (called the “point at infinity”) denote any object outside of X and let X̂ = X ∪ {∞}. Define U ⊆ X̂ to be open in X̂ if and only if either U ⊆ X and U is open in X or U = X̂ −C where C is a compact subset of X. It is easy to check that this gives a topology for X and that X̂ with this topology is a compactification of X. Some easy examples are:

1. If X = (0, 1] = {x ∈ IR : 0 < x ≤ 1}, then X̂ = [0, 1].

2. If X = IR, which is homeomorphic to the open interval (0, 1), then X̂ is homeomorphic to S^{1} = {(x, y) ∈ IR^{2} : x^{2} + y^{2} = 1}.

3. If X is an infinite set with the discrete topology, then every neighborhood of ∞ in X̂ contains all but a finite number of points of X.

4. The Stone-Čech compactification of X

A topological space X is completely regular if single point sets are closed in X and if for each point x_{0} in X and each closed subset A of X which does not contain x_{0}, there is a continuous function f : X → [0, 1] such that f(x_{0}) = 1 and f(A) = {0}. It is well-known that a topological space X has a compactification if and only if X is completely regular. (See, e.g., ([1], corollary 2.3, p. 237)). It should be pointed out here that a locally compact Hausdorff space is completely regular because its one-point compactification, X̂, is compact and Hausdorff, hence X̂ is normal, hence completely regular, and every subspace of a completely regular space is completely regular.

A basic question which arises in studying compactifications of a space X is the following: If Y is a compactification of X, under what conditions can a continuous real-valued function f defined on X be continuously extended to Y? Obviously f must be bounded, because f will carry the compact space Y into IR, and compact subsets of IR must be bounded. But being bounded is not enough. A standard example is f(x) = sin(\frac{1}{x}) defined on (0, 1]. Historically, this problem of continuously extending any bounded, continuous, real-valued function defined on X motivated the development of the Stone-Čech compactification of X, denoted β(X), which we now describe.

Let X be a completely regular space and let {f_{α}}_{α∈J} be the collection of all bounded, continuous, real-valued functions defined on X, indexed by some set J. For each α ∈ J, choose the closed interval I_{α} in IR to be

I_{α} = [glb{f_{α}(X)}, lub{f_{α}(X)}].

Then define h : X → Π_{α∈J}I_{α} by h(x) = (f_{α}(x))_{α∈J}. By the Tychonoff theorem, ΠI_{α} is compact. Since X is completely regular, the collection {f_{α}}_{α∈J} separates points from closed sets in X. Thus by ([1], theorem 4.2, p. 220), h is an imbedding. We define β(X), the Stone-Čech compactification of X, to be the closure of h(X) in ΠI_{α}. See ([1], theorem 3.1, p. 241) for a proof that every bounded, continuous, real-valued function on X is uniquely extendable to β(X).

To show that β(X) ≠ X̂ for a particular space X, we need only exhibit one bounded, continuous, real-valued function f on X which cannot be continuously extended to X̂. If X = (0, 1] or (0, 1), let f(x) = sin(\frac{1}{x}). If X is an infinite set with the discrete topology, let f be any function which takes on different values on two different infinite subsets of X. But there are spaces for which β(X) = X̂, and we introduce such an example in the next section.

5. S_{Ω} and S̄_{Ω}

Let X be an ordered set and α ∈ X. The subset S_{α} = {x ∈ X : x < α} is called a section of X by α. See ([1], corollary, p. 66) for a proof that there exists an uncountable well-ordered set, Y, at least one section of which is uncountable. Consider the subset of Y consisting of those α ∈ Y for which the section S_{α} is uncountable and let Ω denote the smallest such element of Y. Then S_{Ω} is a wellordered set which is uncountable, but every section of S_{Ω} is countable. Munkres also proves ([1] corollary 10.3, p. 67) that every countable subset of S_{Ω} has an upper bound (and therefore a least upper bound) in S_{Ω}.

If α_{0} denotes the smallest element of S_{Ω}, then S_{Ω} is often written as [α_{0}, Ω), and S_{Ω} ∪ {Ω} is written [α_{0}, Ω] and is denoted by S̄_{Ω}. We next show that Ŝ_{Ω} = β(S_{Ω}) = S̄_{Ω} by proving the following theorems:

Theorem 1. Ŝ_{Ω} = S̄_{Ω}.

Proof: First we must show that S̄_{Ω} is compact. Note that {α_{0}} is a nonempty compact subset of S_{Ω}. Let α = lub{x ∈ S̄_{Ω} : S_{x} is compact }. Certainly α cannot be the immediate predecessor of an element β ∈ S̄_{Ω}. But the only element of S̄_{Ω} with no immediate successor is Ω. Thus α = Ω and S̄_{Ω} is compact. Since S̄_{Ω} − S_{Ω} = {Ω}, S̄_{Ω} = Ŝ_{Ω}.

Theorem 2. Every continuous, real-valued function defined on S_{Ω} is eventually constant.

Proof: Suppose there is an ϵ > 0 such that for every α ∈ S_{Ω} there is a β > α such that |f(β) − f(α)| ≥ ϵ. Then for every positive integer n there is an α_{n} ∈ S_{Ω} so that α_{n} > α_{n−1} and |f(α_{n}) − f(α_{n−1})| ≥ ϵ. Let A = {α_{1}, α_{2}, ⋯ } ⊆ S_{Ω} and let α = lub(A) ∈ S_{Ω}. Since f is continuous at α, f(α) should equal lim_{n→∞}f(α_{n}). But lim_{n→∞}f(α_{n}) does not exist, so there cannot be such an ϵ. Hence, for every ϵ > 0, there is an α ∈ S_{Ω} such that |f(β) − f(α)| < ϵ for all β > α.

Now, for every positive integer n, choose γ_{n} ∈ S_{Ω} so that for every β > γ_{n}, |f(β) − f(γ_{n})| < \frac{1}{n}. Let B = \{γ_{1}, γ_{2}, ⋯ \} ⊆ S_{Ω} and let γ = lub(B). It is easy to see that if β > γ, then |f(β) − f(γ)| < \frac{1}{n} for every n, so f(β) = f(γ); i.e., f is constant after β.

It follows immediately from Theorem 2 that β(S_{Ω}) = S̄_{Ω} = Ŝ_{Ω}. It should also be noted that essentially the same proof yields the following generalization of the previous theorem:

Theorem 3. If (Y, d) is any metric space, then every real-valued function f : S_{Ω} → Y is eventually constant.

6. Comparing the Stone-Čech and one-point compactifications

Clearly if X is a locally compact Hausdorff space satisfying that every continuous, real-valued function defined on X is eventually constant (i.e., constant off of some compact subset of X), then β(X) = X̂. It is natural to ask if there are conditions on X which guarantee that β(X) ≠ X̂. An answer is given in the following theorem, which we then generalize in theorem 5. [Note that ClX̂(A) denotes the closure of A in X̂.]

Theorem 4. If X is locally compact, Hausdorff and normal and if there is a countable subset A = {x_{1}, x_{2}, … } of X such that ClX̂(A) = A ∪ {∞}, then β(X) ≠ X̂.

Proof: By assumption, the point at infinity is the only limit point of A in X̂ and therefore A is closed in X. Write A = B ∪ C, where B = {x_{1}, x_{3}, x_{5}, … } and C = {x_{2}, x_{4}, x_{6}, … }. Since X is normal and B and C are disjoint closed subsets of X, the Urysohn lemma implies there is a continuous f : X → [0, 1] such that f(B) = {0} and f(C) = {1}. This f is clearly not continuously extendable to X̂ and thus β(X) ≠ X̂.

An immediate consequence of the previous theorem is that if X̂ is metrizable, or if X can be written as the countable union of compact sets (e.g., IR = ∪_{n=1}^{∞}[−n, n]), then β(X) ≠ X̂. It should be pointed out that the converse of the theorem is false. For example, if X is the disjoint union of two copies of S_{Ω}, then the “point at infinity” in X̂ will be a point (denoted Ω) joining both copies of S_{Ω} at their “ends”. If f is defined from X into the reals by f(x) = 0 for all x in one of the copies of S_{Ω}, and f(x) = 1 for all x in the other, then certainly f cannot be continuously extended to X̂, and hence β(X) ≠ X̂. But there is no (countable) sequence of points in X̂ which can converge to Ω.

The requirement that X be normal is necessary in order to apply the Urysohn lemma. The following lemma allows us to “weaken” that hypothesis to requiring that X be Lindelöf. [Note that a space X is Lindelöf if every open covering of X has a countable subcovering.]

Lemma 1. If X is locally compact, Hausdorff and Lindelöf, then X is normal.

Proof: It is a standard sequence of exercises (see [1], p. 205, exercises 6,7) to show that every locally compact, Hausdorff space is regular and that every regular, Lindelöf space is normal.

Theorem 5. If X is locally compact, Hausdorff and Lindelöf, then β(X) ≠ X̂.

Proof: By theorem 4, we need only show that there is a countable subset A of X such that ClX̂(A) = A ∪ {∞}. Since X is locally compact, for each x ∈ X, there is an open neighborhood U_{x} of x such that Ū_{x} = Cl_{X}(U_{x}) is compact. Then U = {U_{x} : x ∈ X} is an open cover of X. Since X is Lindelöf, there is a countable subcover, say {U_{x_{1}}, U_{x_{2}}, … }. For each positive integer n, let V_{n} = ∪_{i=1}^{n}U_{x_{i}} (hence, of course, V̄_{n} = Cl_{X}(V_{n}), which is compact) and choose y_{n} ∈ X − V̄_{n}. Let A = {y_{1}, y_{2}, … }. If x ∈ X − A, then x ∈ U_{x_{n}} for some n, so U_{x_{n}} is a neighborhood of x which does not contain any of y_{n}, y_{n+1}, y_{n+2}, . . .. Since X is Hausdorff, we can find a neighborhood of x disjoint from A. Thus x ∉ Cl_{X}(A). Now, if C is a compact subset of X, (so that X − C is a neighborhood of ∞ in X̂), then there is a V_{n} such that C ⊆ V_{n}. Thus {y_{n}, y_{n+1}, y_{n+2}, … } ⊆ (X̂ − C); i.e., every neighborhood of ∞ in X̂, contains points of A. Hence ∞ ∈ ClX̂(A), or ClX̂(A) = A ∪ {∞}.

In general, trying to visualize or understand the Stone-Čech compactification of a topological space can be mind-boggling if not impossible. We have seen that this is not the case for S_{Ω}. Next we calculate β(S_{Ω} × X) where X is a compact space. First consider C(X, IR) = the collection of all continuous f : X → IR.

Lemma 2. d : C(X, IR)×C(X, IR) → IR defined by d(f, g) = lub_{x∈X}{|f(x)−g(x)|} defines a metric on C(X, IR).

Proof: Clearly d(f, g) ≥ 0 and d(f, g) = 0 if and only if f = g. Also, it is clear that d(f, g) = d(g, f). Suppose f, g, h ∈ C(X, IR). For any x ∈ X, |f(x) − g(x)| = |f(x) − h(x) + h(x) − g(x)| ≤ |f(x) − h(x)| + |h(x) − g(x)| ≤ lub{|f(x) − h(x)|} + lub{|h(x) − g(x)|} = d(f, h) + d(h, g). Since this holds for any x ∈ X, we have lub{|f(x) − g(x)|} ≤ d(f, h) + d(h, g); or, d(f, g) ≤ d(f, h) + d(h, g).

Theorem 6. If X is compact, then β(S_{Ω} × X) = S̄_{Ω} × X.

Proof: Let f : S_{Ω} × X → IR be a bounded, continuous function. Define f̂: S_{Ω} → C(X, IR) by f̂(α)(x) = f(α, x). By ([1], corollary 5.4, p. 287), f̂ is continuous. Since C(X, IR) is a metric space, theorem 3 implies that f̂ is eventually constant and hence continuously extendable to F̂ : S̄_{Ω} → C(X, IR). Hence F : S̄_{Ω} × X → IR defined by F (α, x) = F̂(α)(x) is a continuous extension of f, and therefore β(S_{Ω} × X) = S̄_{Ω} × X.

Note that β(S_{Ω} × X) = S̄_{Ω} × X is constructed from S_{Ω} × X by “adding” an entire copy of X, not just a single point. Thus the one-point compactification of S_{Ω} × X is not β(S_{Ω} × X).

Next we describe a new example of a space X for which X̂ = β(X). Let IR denote the reals, β(IR) the Stone-Čech compactification of IR, and choose x ∈ β(IR) − IR. Let X = β(IR) − {x}. If f is any bounded, continuous, real-valued function on X, then f|_{IR}, the restriction of f to IR, has a unique extension to β(IR), say f̃. But since Cl_{X}(IR) = X, f|_{IR} has at most one continuous extension to X (see [1], Lemma 3.2, p. 241). Since both f and f̃|_{X} are continuous extensions of f|_{IR}, they must be identical. Thus f : X → IR has a continuous extension, f̃, to β(IR) and so β(X) = β(IR). But clearly X̂ = X ∪ {x} = β(IR), so X̂ = β(X). This construction can be generalized as:

Theorem 7. Let X be a locally compact Hausdorff space, x ∈ β(X)−X, and Y = β(X) − {x}. Then Ŷ = β(Y).

References

[1] Munkres, James R., Topology, A First Course, Prentice-Hall, 1975.

Jay Blankespoor: Undergraduate at Calvin College

   E-mail address: jblank48@calvin.edu

John Krueger: Undergraduate at Hope College

   E-mail address: krueger@cs.hope.edu

Sponsor: John Richard Stoughton, Department of Mathematics, Hope College, Holland, MI 49423

   E-mail address: stoughton@math.hope.edu