One way to study a general class of mathematical objects is to embed them into a more structured class of mathematical objects; for instance, one could study manifolds by embedding them into Euclidean spaces. In these (optional) notes we study two (related) embedding theorems for topological spaces:

— 1. The Stone-Čech compactification —

Observe that any dense open subset of a compact Hausdorff space is automatically a locally compact Hausdorff space. We now study the reverse concept:

Definition 1 A compactification of a locally compact Hausdorff space {X} is an embedding {\iota: X \rightarrow \overline{X}} (i.e. a homeomorphism between {X} and {\iota(X)}) into a compact Hausdorff space {\overline{X}} such that the image {\iota(X)} of {X} is an open dense subset of {\overline{X}}. We will often abuse notation and refer to {\overline{X}} as the compactification rather than the embedding {\iota: X \rightarrow \overline{X}}, when the embedding is obvious from context.

One compactification {\iota: X \rightarrow \overline{X}} is finer than another {\iota': X \rightarrow \overline{X}'} (or {\iota': X \rightarrow \overline{X}'} is coarser than {\iota: X \rightarrow \overline{X}}) if there exists a continuous map {\pi: \overline{X}' \rightarrow \overline{X}} such that {\iota = \pi \circ \iota'}; notice that this map must be surjective and unique, by the open dense nature of {\iota(X)}. Two compactifications are equivalent if they are both finer than each other.

Example 1 Any compact set can be its own compactification. The real line {{\mathbb R}} can be compactified into {[-\pi/2,\pi/2]} by using the arctan function as the embedding, or (equivalently) by embedding it into the extended real line {[-\infty,\infty]}. It can also be compactified into the unit circle {\{ (x,y) \in {\mathbb R}^2: x^2 + y^2 = 1 \}} by using the stereographic projection {x \mapsto (\frac{2x}{1+x^2}, \frac{x^2-1}{1+x^2})}. Notice that the former embedding is finer than the latter. The plane {{\mathbb R}^2} can similarly be compactified into the unit sphere {\{ (x,y,z) \in {\mathbb R}^2: x^2 + y^2 +z^2 = 1 \}} by the stereographic projection {(x,y) \mapsto (\frac{2x}{1+x^2+y^2}, \frac{2y}{1+x^2+y^2}, \frac{x^2+y^2-1}{1+x^2+y^2})}.

Exercise 1 Let {X} be a locally compact Hausdorff space {X} that is not compact. Define the one-point compactification {X \cup \{\infty\}} by adjoining one point {\infty} to {X}, with the topology generated by the open sets of {X}, and the complement (in {X \cup \{\infty\}}) of the compact sets in {X}. Show that {X \cup \{\infty\}} (with the obvious embedding map) is a compactification of {X}. Show that the one-point compactification is coarser than any other compactification of {X}.

We now consider the opposite extreme to the one-point compactification:

Definition 2 Let {X} be a locally compact Hausdorff space. A Stone-Čech compactification {\beta X} of {X} is defined as the finest compactification of {X}, i.e. the compactification of {X} which is finer than every other compactification of {X}.

It is clear that the Stone-Čech compactification, if it exists, is unique up to isomorphism, and so one often abuses notation by referring to the Stone-Čech compactification. The existence of the compactification can be established by Zorn’s lemma (see these lecture notes of mine from last year). We shall shortly give several other constructions of the compactification. (All constructions, however, rely at some point on the axiom of choice, or a related axiom.)

The Stone-Čech compactification obeys a useful functorial property:

Exercise 2 Let {X, Y} be locally compact Hausdorff spaces, with Stone-Čech compactifications {\beta X, \beta Y}. Show that every continuous map {f: X \rightarrow Y} has a unique continuous extension {\beta f: \beta X \rightarrow \beta Y}. (Hint: uniqueness is easy; for existence, look at the closure of the graph {\{ (x,f(x)): x \in X \}} in {\beta X \times \beta Y}, which compactifies {X} and thus cannot be strictly finer than {\beta X}.) In the converse direction, if {\overline{X}} is a compactification of {X} such that every continuous map {f: X \rightarrow K} into a compact space can be extended continuously to {\overline{X}}, show that {\overline{X}} is the Stone-Čech compactification.

Example 2 From the above exercise, we can define limits {\lim_{x \rightarrow p} f(x) := \beta f(p)} for any bounded continuous function on {X} and any {p \in \beta X}. But one for coarser compactifications, one can only take limits for special types of bounded continuous functions; for instance, using the one-point compactification of {{\mathbb R}}, {\lim_{x \rightarrow \infty} f(x)} need not exist for a bounded continuous function {f: {\mathbb R} \rightarrow {\mathbb R}}, e.g. {\lim_{x \rightarrow \infty} \sin(x)} or {\lim_{x \rightarrow \infty} \arctan(x)} do not exist. The finer the compactification, the more limits can be defined; for instance the two point compactification {[-\infty,+\infty]} of {{\mathbb R}} allows one to define the limits {\lim_{x \rightarrow +\infty} f(x)} and {\lim_{x \rightarrow -\infty} f(x)} for some additional functions {f} (e.g. {\lim_{x \rightarrow \pm \infty} \arctan(x)} is well-defined); and the Stone-Čech compactification is the only compactification which allows one to take limits for any bounded continuous function (e.g. {\lim_{x \rightarrow p} \sin(x)} is well-defined for all {p \in \beta {\mathbb R}}).

Now we turn to the issue of actually constructing the Stone-Čech compactifications.

Exercise 3 Let {X} be a locally compact Hausdorff space. Let {C(X \rightarrow [0,1])} be the space of continuous functions from {X} to the unit interval, let {Q := [0,1]^{C(X \rightarrow [0,1])}} be the space of tuples {(y_f)_{f \in C(X \rightarrow [0,1])}} taking values in the unit interval, with the product topology, and let {\iota: X \rightarrow Q} be the Gelfand transform {\iota(x) := (f(x))_{f \in C(X \rightarrow [0,1])}}, and let {\beta X} be the closure of {\iota X} in {Q}.

  • Show that {\beta X} is a compactification of {X}. ({\emph Hint}: Use Urysohn’s lemma and Tychonoff’s theorem.)
  • Show that {\beta X} is the Stone-Čech compactification of {X}. ({\emph Hint}: If {\overline{X}} is any other compactification of {X}, we can identify {C(\overline{X} \rightarrow [0,1])} as a subset of {C(X \rightarrow [0,1])}, and then project {Q} to {[0,1]^{C(\overline{X} \rightarrow [0,1])}}. Meanwhile, we can embed {\overline{X}} inside {[0,1]^{C(\overline{X} \rightarrow [0,1])}} by the Gelfand transform.)

Exercise 4 Let {X} be a discrete topological space, let {2^X} be the Boolean algebra of all subsets of {X}. By Stone’s representation theorem (Theorem 1 from Notes 1), {2^X} is isomorphic to the clopen algebra of a Stone space {\beta X}.

  • Show that {\beta X} is a compactification of {X}.
  • Show that {\beta X} is the Stone-Čech compactification of {X}.
  • Identify {\beta X} with the space of ultrafilters on {X}. (See this post for further discussion of ultrafilters, and this post for further discussion of the relationship of ultrafilters to the Stone-Čech compactification.)

Exercise 5 Let {X} be a locally compact Hausdorff space, and let {BC(X \rightarrow {\mathbb C})} be the space of bounded continuous complex-valued functions on {X}.

  • Show that {BC(X \rightarrow {\mathbb C})} is a unital commutative {C^*}-algebra (see Section 4 of Notes 12).
  • By the commutative Gelfand-Naimark theorem (Theorem 14 of Notes 12), {BC(X \rightarrow {\mathbb C})} is isomorphic as a unital {C^*}-algebra to {C(\beta X \rightarrow {\mathbb C})} for some compact Hausdorff space {\beta X} (which is in fact the spectrum of {BC(X \rightarrow {\mathbb C})}. Show that {\beta X} is the Stone-Čech compactification of {X}.
  • More generally, show that given any other compactification {\overline{X}} of {X}, that {C(\overline{X} \rightarrow {\mathbb C})} is isomorphic as a unital {C^*}-algebra to a subalgebra of {BC(X \rightarrow {\mathbb C})} that contains {{\mathbb C} \oplus C_0(X \rightarrow {\mathbb C})} (the space of continuous functions from {X} to {{\mathbb C}} that converge to a limit at {\infty}), with {\overline{X}} as the spectrum of this algebra; thus we have a canonical identification between compactifications and {C^*}-algebras between {BC(X \rightarrow {\mathbb C})} and {{\mathbb C} \oplus C_0(X \rightarrow {\mathbb C})}, which correspond to the Stone-Čech compactification and one-point compactification respectively.

Exercise 6 Let {X} be a locally compact Hausdorff space. Show that the dual {BC(X \rightarrow {\mathbb R})^*} of {BC(X \rightarrow {\mathbb R})} is isomorphic as a Banach space to the space {M(\beta X)} of real signed Radon measures on the Stone-Čech compactification {\beta X}, and similarly in the complex case. In particular, conclude that {\ell^\infty({\mathbb N})^* \equiv M(\beta {\mathbb N})}.

Remark 1 The Stone-Čech compactification can be extended from locally compact Hausdorff spaces to the slightly larger class of Tychonoff spaces, which are those Hausdorff spaces {X} with the property that any closed set {K \subset X} and point {x} not in {K} can be separated by a continuous function {f \in C(X \rightarrow {\mathbb R})} which equals {1} on {K} and zero on {x}. This compactification can be constructed by a modification of the argument used to establish Exercise 3. However, in this case the space {X} is merely dense in its compactification {\beta X}, rather than open and dense.

Remark 2 A cautionary note: in general, the Stone-Čech compactification is almost never sequentially compact. For instance, it is not hard to show that {{\mathbb N}} is sequentially closed in {\beta {\mathbb N}}. In particular, these compactifications are usually not metrisable.

— 2. Urysohn’s metrisation theorem —

Recall that a topological space is metrisable if there exists a metric on that space which generates the topology. There are various necessary conditions for metrisability. For instance, we have seen that metric spaces must be normal and Hausdorff. In the converse direction, we have

Theorem 3 (Urysohn’s metrisation theorem) Let {X} be a normal Hausdorff space which is second countable. Then {X} is metrisable.

Proof: (Sketch) This will be a variant of the argument in Exercise 3, but with a countable family of continuous functions in place of {C(X \rightarrow [0,1])}.

Let {U_1, U_2, \ldots} be a countable base for {X}. If {U_i, U_j} are in this base with {\overline{U_i} \subset U_j}, we can apply Urysohn’s lemma and find a continuous function {f_{ij}: X \rightarrow [0,1]} which equals {1} on {\overline{U_i}} and vanishes outside of {U_j}. Let {{\mathcal F}} be the collection of all such functions; this is a countable family. We can then embed {X} in {[0,1]^{{\mathcal F}}} using the Gelfand transform {x \mapsto (f(x))_{f \in {\mathcal F}}}. By modifying the proof of Exercise 3 one can show that this is an embedding. On the other hand, {[0,1]^{{\mathcal F}}} is a countable product of metric spaces and is thus metrisable (e.g. by enumerating {{\mathcal F}} as {f_1, f_2, \ldots} and using the metric {d( (x_n)_{f_n \in {\mathcal F}}, (y_n)_{f_n \in {\mathcal F}} ) := \sum_{n=1}^\infty 2^{-n} |x_n-y_n|}). Since a subspace of a metrisable space is clearly also metrisable, the claim follows. \Box

Recalling that compact metric spaces are second countable (Lemma 4 of Notes 10), thus we have

Corollary 4 A compact Hausdorff space is metrisable if and only if it is second countable.

Of course, non-metrisable compact Hausdorff spaces exist; {\beta {\mathbb N}} is a standard example. Uncountable products of non-trivial compact metric spaces, such as {\{0,1\}}, are always non-metrisable. Indeed, we already saw in Notes 10 that {\{0,1\}^X} is compact but not sequentially compact (and thus not metrisable) when {X} has the cardinality of the continuum; one can use the first uncountable ordinal to achieve a similar result for any uncountable {X}, and then by embedding one can obtain non-metrisability for any uncountable product of non-trivial compact metric spaces, thus complementing the metrisability of countable products of such spaces. Conversely, there also exist metrisable spaces which are not second countable (e.g. uncountable discrete spaces). So Urysohn’s metrisation theorem does not completely classify the metrisable spaces, however it already covers a large number of interesting cases.