This is a continuation of an introduction of Stone-Cech compactification started in two previous posts (first post: A Beginning Look at Stone-Cech Compactification; second post: Two Characterizations of Stone-Cech Compactification). In this post, we present another characterization of the Stone-Cech compactification, that is, for any completely regular space , is -embedded in its Stone-Cech compactification and that any compactification of in which is -embedded must be . In other words, this property of -embedding is unique to Stone-Cech compactification. We prove the following two theorems (U3 has two versions).
The links for other posts on Stone-Cech compactification can be found toward the end of this post.
Definition. Let be a space. Let . The subspace is -embedded in if every bounded continuous function is extendable to a continuous .
Let be a completely regular space. The space is -embedded in its Stone-Cech compactification .
Let be a completely regular space. Let . Let be a compactification of such that each continuous can be extended to a continuous . Then must be .
If is any compactification of that satisfies the property in Theorem C3 (i.e., is -embedded in ), then must be .
Two other characterizations of are proved in the previous post (Two Characterizations of Stone-Cech Compactification).
Let be a completely regular space. Let be a continuous function from into a compact Hausdorff space . Then there is a continuous such that .
If is any compactification of that satisfies condition in Theorem C1, then must be equivalent to .
Let be a completely regular space. Among all compactifications of the space , the Stone-Cech compactification of the space is the largest compactification.
The property in Theorem C2 is unique to . That is, if is a compactification of , then must be equivalent to .
The C theorems and the U theorems are a great tool to determine whether a given compactification is . Whenever a compactification of a space satisfies the property belonging to a C theorem, based on the corresponding U theorem, we know that this compactification must be . For example, any compactification that satisfies the function extension property in Theorem C1 must be . Th -embedding property in Theorem C3 and Theorem U3 (both versions) is also a function extension property much like that in Theorems C1 and U1, but is easier to use. The reason being that we only need to extend a smaller class of continuous functions (i.e., to check whether functions from into can be extended), rather than checking all continuous functions from to arbitrary compact spaces. As the following example below about illustrates that the -embedding in Theorem C3 and U3.1 can be used to describe explicitly.
Proving Theorem U3.1 and Theorem U3.2
Let be a space. Let be a subspace of . Recall that is -embedded in if every bounded continuous function can be extended to a continuous .
Any bounded continuous function can be regarded as where is some closed and bounded interval. The -embedding property in Theorem C3 is a function extension property like the one in Theorem C1, except that it deals with function from into a specific type of compact spaces , namely the closed and bounded intervals in . Theorem C3 is a corollary of Theorem C1 (see below). So we only need to prove Theorem U3.1 and Theorem U3.2. Theorem U3.2 is a corollary of Theorem U3.1.
Proof of Theorem U3.1
By Theorem C2, we have . So we only need to show . To this end, we need to produce a continuous function such that .
Let be the set of all continuous functions from into . For each , let . Recall that is embedded in the cube by the mapping . For each , let be the projection map from this cube into .
Each can be expressed as . Thus by assumption, each can be extended by . Now define by the following:
For each , such that
For each , we have . Note that agrees with on since extends . So we have where for each . On the other hand, by definition of , we have where for each . Thus we have and the following:
It is straightforward to verify that is continuous. Note that is dense in . Since is continuous, is dense in . Thus we have:
Putting and together, we have the following:
Thus we can describe the map as . As noted before, we have . Thus .
Proof of Theorem U3.2
Suppose is a compactification of such that is -embedded in . Then every bounded continuous can be extended to where is some closed and bounded interval containing the range. In particular, this means every continuous can be extended. By Theorem U3.1, we have .
This is one example where we can use -embedding to describe explicitly.
Let be the first uncountable ordinal. Let be the successor ordinal of (i.e. with one additional point at the end). Consider and as topological spaces with the order topology derived from the well ordering of the ordinals. The space is a compactification of . In fact is the one-point compactification of .
It is well known that every continuous real-valued function on is bounded (note that here is countably compact and hence pseudocompact). Furthermore, every continuous real-valued function on is eventually constant. This means that if is continuous, for some , is constant on the final segment (see result B in The First Uncountable Ordinal). As a result, every continuous bounded real-valued function can be extended to a continuous . Then according to Theorem U3.2, .
Blog Posts on Stone-Cech Compactification
Post #0: Embedding Completely Regular Spaces into a Cube
Post #3 (this post): C*-Embedding Property and Stone-Cech Compactification
- Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
- Willard, S., General Topology, Addison-Wesley Publishing Company, 1970.