In this post we prove a lemma that is a great tool for working with product spaces of separable metrizable spaces. As an application of the lemma, we give an alternative proof for showing the non-normality of the product space of uncountably many copies of the discrete space of the non-negative integers.
Consider the product space where each is a separable and metrizable space. The lemma we discuss here is a tool that can shed some light on normality of dense subspaces of the product space . The lemma is stated in two equivalent forms (Lemma 1 and Lemma 2).
Before stating the lemmas, let’s fix some notations. For any , the map is the natural projection from the full product to the subproduct . The standard basic open sets in the product space are of the form where for all but finitely many . We use to denote the set of finitely many such that .
Given a space , and given , the sets and are separated if .
Let be a product of separable metrizable spaces. Let be a dense subspace of . For any sets , the following two conditions are equivalent:
- There exist disjoint open subsets and of such that and .
- There exists a countable such that the sets and are separated in the space .
Let be a product of separable metrizable spaces. Let be a dense subspace of . Then is normal if and only if for each pair of disjoint closed subsets and of , there exists a countable such that and are separated in .
If Lemma 1 holds, it is clear that Lemma 2 holds. We prove Lemma 1. The lemmas indicate that to separate disjoint sets in the full product, it suffices to separate in a countable subproduct. In this sense normality in dense subspaces of a product of separable metrizable spaces only depends on countably many coordinates.
This lemma seems to have been around for a long time. We cannot find any reference of this lemma in Engelking’s topology textbook (see ). We found three references. One is Corson’s paper (see ), in which the lemma is mentioned in relation to the non-normality of and is attributed to a paper of M. Bockstein in 1948. Another is a paper of Baturov (see ), in which the lemma is used to prove a theorem about normality in dense subspace of where is a separable metric space. In  the lemma is attributed to Uspenskii. Another reference is Arkhangelskii’s book on function space (see Lemma I.6.1 on p. 43 in ) where the lemma is used in proving some facts about normality in function spaces .
Proof of Lemma 1
Let and be disjoint open subsets of with and . Let and be open subsets of such that and . Since is dense in , .
Let be a maximal pairwise disjoint collection of standard basic open sets, each of which is a subset of . Let be a maximal pairwise disjoint collection of standard basic open sets, each of which is a subset of . These two collections can be obtained using a Zorn lemma argument. The product space has the countable chain condition since it is a product of separable spaces. So both and are countable. Let be the union of finite sets each one of which is a where . The set is countable too.
Let and . Note that . We have the following observations:
The above observations lead to the following observations:
implying that . Both and are open subsets of and are dense in , respectively.
We claim that . Suppose that . Then contains a point of , say . With , for some where . Note that . Thus . On the other hand, implies that for some . It follows that , a contradiction. Therefore .
We have and . This implies that and (closure in ). Then and are separated in as well. This concludes the proof for the direction.
Suppose that is countable such that and are separated in the space . Note that and . Then we have the following:
Consider . The space is an open subspace of . Furthermore, is a subspace of , which is a separable and metrizable space. Thus the space is metrizable and hence normal.
For , let denote the closure of in the space . Note that and are disjoint and closed sets in . Let and be disjoint open subsets of such that and . Then and are disjoint open subsets of such that and .
The proof of Lemma 1 does not need the full strength of separable metric in each factor of the product space. The above proof only makes two assumptions about the product space: the product space has the countable chain condition (CCC) and that any countable subproduct is normal, i.e., is normal for any countable .
As an application of the above lemma, we give another proof of the non-normality of the product space of uncountably many copies of the discrete space of the non-negative integers. See this post for a version of A. H. Stone’s original proof.
Let be the set of all nonnegative integers and let be the first uncountable ordinal (i.e. the set of all countable ordinals). We provide an alternative proof that is not normal. In A. H. Stone’s proof, the following disjoint closed sets cannot be separated in :
We can also use Lemma 1 to show that and cannot be separated. Note that for each countable , the sets and have non-empty intersection. Hence they cannot be separated in . By Lemma 1, and cannot be separated in the full product space .
To see that , choose a function such that . Let be defined by for all . Then .
- Arkhangelskii, A. V., Topological Function Spaces, Mathematics and Its Applications Series, Kluwer Academic Publishers, Dordrecht, 1992.
- Baturov, D. P., Normality in dense subspaces of products, Topology Appl., 36, 111-116, 1990.
- Corson, H. H., Normality in subsets of product spaces, Amer. J. Math., 81, 785-796, 1959.
- Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.