This is a continuation of a discussion on completely regular spaces (continuing from these two posts: Completely Regular Spaces and Pseudocompact Spaces, Completely Regular Spaces and Function Spaces). This post gives another reason (one of the most important ones) why the class of completely regular spaces occupies a central place in general topology, which is that completely regular spaces are precisely the spaces that can be embedded in a cube (the product of copies of the unit interval). This theorem was proved by Tychonoff in 1930 . The tool that makes this theorem possible also allowed Stone and Cech in 1937 to construct for any completely regular space , a compact Hausdorff space that contains as a dense set ( is called the Stone-Cech compactification of ). In this post we discuss the role played by complete regularity in this construction. We discuss the following theorem.
Let be a space. Then is completely regular if and only if is homeomorphic to a subspace of a cube.
Original articles are ,  and . The Stone-Cech compactification and the related concepts are classic topics that are covered in standard texts (see  and ).
Completely Regular Spaces
A space is said to be completely regular if is a space and for each and for each closed subset of with , there is a continuous function such that and . Note that the axiom and the existence of the continuous function imply the axiom, which is equivalent to the property that single points are closed sets. Completely regular spaces are also called Tychonoff spaces.
The Evaluation Map
The evaluation map can be defined in a more general setting. We define it here just to deal with the task at hand, namely to discuss Theorem 1.
Let be a completely regular space. Let be the unit interval in the real line . A cube is of the form , i.e., the product of many copies of where is some cardinal. Let be the set of all continuous real-valued functions defined on the space . Consider the product space where each . This is the cube where is embedded as a subspace. We can represent each point in the cube as a function or as a sequence such that each term (or coordinate) .
The key to embed into a cube is through the evaluation map, which is a map from into the product space . Thus we have:
We now define the map . For each , such that for each . In other words, is the point in the product space whose coordinate is .
We show that because is completely regular, the evaluation map is a homeomorphism. We show the following:
- The map is continuous.
- The map is one-to-one.
- The map is continuous.
The continuity of the map follows from the fact that each is continuous. The map being one-to-one follows from the fact that for each pair with , there is an such that .
We now show , the inverse of , is continuous. This is where must be completely regular. Let be open in such that . Since is completely regular, there is a continuous such that and . Let where for all and . Then is open in and . We claim that . To see this, pick . Note that since . If , then . Thus we must have .
The above discussion shows that any completely regular space is homeomorphic to a subspace of the product space where is the cardinality of . This establishes one direction of Theorem 1. The other direction is clear. Note that any cube is a compact Hausdorff space and any subspace of a compact Hausdorff space is completely regular.
The Stone-Cech Compactification
Let be a completely regular space. Let be the evaluation as defined in above. According to the above discussion is a homeomorphism. Hence is a topological copy of as a subspace of the product space . Consider where the closure is taken in the product space. The Stone-Cech compactification of is denoted by and is defined to be this closure .
See  and  for more information.
- Cech, E., On bicompact spaces, Ann. Math. (2) 38, 823-844, 1937.
- Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
- Steen, L. A., Seebach, J. A., Counterexamples in Topology, Dover Publications, Inc., New York, 1995.
- Stone, M. H., Applications of the Theorey of Boolean Rings to General Topology, Trans. Amer. Math. Soc., 41, 375-481, 1937.
- Tychonoff, A., Uber die topologische von Raumen, Math., Ann., 102, 544-561, 1930.
- Willard, S., General Topology, Addison-Wesley Publishing Company, 1970.