It follows from the Tychonoff Theorem that if a topological space is a compact Hausdorff space, then the product space is normal for any cardinal number . The converse is also true. If is normal for any cardinal number , then must be compact. This is a theorem that is due to Noble (see ). The proof in  is a corollary resulted from a long chain of previous results of Noble and others. Many authors had produced simpler and more direct proofs of Noble’s theorem (e.g. ,  and ). All these more direct proofs make use of the fact that the product space is not normal (due to A. H. Stone). All of them except  make use of other strong topological results in order to derive Noble’s theorem. In , Engelking established Noble’s theorem by an elementary proof. In this post, we present the proof in  in full details. Noble’s theorem is also given in Engelking’s textbook as an exercise (see 3.12.15 on p. 233 in ).
Before proceeding to the main theorem, let’s set up some notation for working with the product space . For , the coordinate of is denoted by or . For , the map is the natural projection map. In the product space , standard basic open sets are of the form where for all but finitely many . We use to denote the set of the finite number of where .
If each power of a space is normal, then is compact.
Suppose that is normal for all cardinal numbers . Suppose that is not compact. Then there exists a collection of closed subsets of such that has the finite intersection property but has empty intersection. Let , which is a subspace of the product space where each . We can also denote the product space by where .
Let . Note that is commonly referred to as the diagonal of the product space in question. Both and are closed sets in the product space . Because the collection has empty intersection, and are disjoint closed sets. Since is normal, there exist disjoint open subsets and of such that and .
Let . Let be a basic standard open set with . Let . Then we have . Since has the finite intersection property, choose . Then define such that for all and for all .
Let be a basic standard open set with . Let . By making a smaller open set if necessary, we can have . Then we have . Choose . Then define such that for all and for all .
After this inductive process is completed, we can obtain:
- a sequence of points of ,
- a sequence of finite subsets of the index set ,
- a sequence of points of
such that for each , for all and .
By A. H. Stone’s theorem (Theorem 5 in ), cannot contain a closed copy of (the space of the positive integers with the discrete topology). A proof that is also found in this post. Let . Either is infinite or finite.
Assume that is an infinite set. Then has a limit point , meaning that every open subset of containing contains some different from . For each , define such that
- for all
- for all
It is the case that for all , since agrees with on the finite set . Let such that for all . It follows that is a limit point of . Thus . Since , the open set would have to contain points of . But and are supposed to be disjoint open subsets of the product space . Thus we have a contradiction.
Assume that is a finite set. Then for some , for all . For each , define such that
- for all
- for all
Let such that for all . Then for all . As in Case 1, for all , implying that . This is a contradiction, since and and are supposed to be disjoint.
Both cases lead to a contradiction. Thus if all powers of is normal, must be compact. This completes the proof of the theorem.
- Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
- Engelking, R., An elementary proof of Noble’s theorem on normality of powers, Comment. Math. Univ. Carolinae, 29.4, 677-678, 1988.
- Franklin, S. P., Walker, R. C., Normalit of powers implies compactness, Proc. Amer. Math. Soc., 36, 295-296, 1972.
- Keesling, J., Normality and infinite product spaces, Adv. in. Math., 9, 90-92, 1972.
- Noble, N., Products with closed projections, II, Trans. Amer. Math. Soc., 160, 169-183, 1971.
- Ross, K. A., Stone, A. H. Products of separable spaces, Amer. Math. Monthly, 71, 398-403, 1964.