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.
In this diagram, is a completely regular space and is the Stone-Cech compactification of where is the homeomorphism mapping onto , which is dense in . The function is an arbitrary continuous function where is compact. Then there exists a continuous function such that restricted to is identical to the function . In other words, if we think of as a subset of , any continuous function from to a compact space can be extended to all of . This function extension property is stated in Theorem C1 below.
Let be a completely regular space. Let be a continuous function from into a compact Hausdorff space . Then there is a continuous such that . See Figure 1 above.
If is any compactification of that satisfies condition in Theorem C1, then must be equivalent to .
Theorem C1 is the statement of the extension property described at the beginning. Theorem U1 states that this property is unique to . That is, of all the possible compactifications of , only can satisfy Theorem C1.
For the other characterization, see Theorem C2 and Theorem U2 below.
Defining Stone-Cech Compactification
The definition of is given in this previous post (A Beginning Look at Stone-Cech Compactification) and is repeated here again for the sake of completeness. Let be the set of all continuous functions from into . For each , . The map is defined by:
For each , is the point such that for each (i.e. the coordinate of the point is ).
For the proof that is a homeomorphism, see A Beginning Look at Stone-Cech Compactification. We have the following definition.
Under the map , is the topological copy of within the cube . The Stone-Cech compactification of is defined to be the closure of in the cube , i.e., set .
When there is no ambiguity as to what the space is, the embedding is written as and the compactification is written as (as in Figure 1 above). When more than one space is involved, we use subscripts to distinguish the embeddings, e.g., and .
Proof of Theorem U1
Let be a continuous function from into a compact Hausdorff space . Let be the Stone-Cech compactification of where is the homeomorphic embedding that defines . Since is a completely regular space, it has a Stone-Cech compactification , where is the homeomorphic embedding. We also define a map from into . We have the following diagram.
The desired function will be defined by . The rest of the proof is to define and to show that this definition of makes sense.
To define the function , for each , let such that (i.e. the coordinate of is the coordinate of ). With the definition of , the diagram in Figure 2 commutes, i.e.,
Starting with a point (the upper left corner of the diagram), we can reach the same point in the lower right corner regardless the path we take ( or ). The following shows the derivation.
The other direction:
It is straightforward to verify that the map is continuous. Based on above, note that . The following derivation shows that .
With the above derivation, we now know that the function maps points of to points of . So it makes sense to define . Note that for each , we have:
Then we have and is the desired function.
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 are equivalent. In the process, we produce another characterization of Stone-Cech compactification (see Theorem C2 and Theorem U2 below).
Let be a completely regular space. A pair is said to be a compactification of the space if is a compact Hausdorff space and is a homeomorphism from into such that is dense in . More informally, a compactification of the space can also be thought of as a compact space containing a topological copy of the space as a dense subspace.
Given a compactification , we use the notation rather than the pair . By saying that is a compactification of , we mean is the compact space where is the homeomorphism embedding onto .
The Stone-Cech compactification construction above is an example of a compactification. There can be more than one compactification of a given space . For example, for , we have the Stone-Cech compactification , which is a subspace of the cube . The circle contains a copy of the real line as a dense subspace, as does the unit interval . Thus both and are also compactifications of . See A Beginning Look at Stone-Cech Compactification for a discussion of these examples.
We say that compactifications and are equivalent (we write ) if there exists a homeomorphism such that . In other words, the following diagram commutes.
Essentially, two compactifications and of are equivalent if there is a homeomorphism between the two and if each is mapped by to itself, i.e., is mapped to .
For a given completely regular space , let be the class of all compactifications of . We define a partial order on . For and , both in , we say that if there is a continuous function such that . See Figure 4 below.
The following theorem ties the partial order to the equivalence relation for compactifications.
Let and be two compactifications of . Then and if and only if .
Proof of Theorem 1
With , there exists continuous such that
With , there exists continuous such that
Applying to , we have . Applying to this result, we have
Note that is a map from into . The equation indicates that when is restricted to , it is the identity map. Thus agrees with the identity map on the dense set . This implies that must agree with the identity map on all of .
Likewise we can see that must equal to the identity map on . So is a homeomorphism and it follows that and are equivalent compactifications of .
This direction is straightforward. Let a homeomorphism that makes and equivalent (as described by Figure 3). Then the map implies and the map implies .
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 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 .
Let be a completely regular space. Among all compactifications of the space , the Stone-Cech compactification of the space is maximal with respect to the partial order .
The property in Theorem C2 is unique to . That is, if, among all compactifications of the space , is maximal with respect to the partial order , then .
Proof Theorem C2
Let be any compactification of . Consider the continuous map . By Theorem C1, can be extended to . In other words, there exists a continuous such that . The existence of the map implies that .
Proof Theorem U2
Let be another maximal compactification of . This implies that . By Theorem C2, we have . By Theorem 1, must be equivalent to .
Proof of Theorem U1
We are now ready to prove Theorem U1.
Proof of Theorem U1
Let be a compactification of that satisfies the extension property in Theorem C1. In light of Theorem C2, we have . So we only need to show . Consider the map . By the assumption that satisfies the extension property in Theorem C1, there exists a continuous function such that . The existence of implies that . By Theorem 1, must be equivalent to .
Blog Posts on Stone-Cech Compactification
Post #0: Embedding Completely Regular Spaces into a Cube
Post #2 (this post): Two Characterizations of Stone-Cech Compactification
- Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
- Willard, S., General Topology, Addison-Wesley Publishing Company, 1970.