The previous post gives a positive result for normality in product space. It shows that the product of a normal countably compact space and a metric space is always normal. In this post, we discuss another positive result, which is the following theorem.
If is a perfectly normal space and is a metric space, then is a perfectly normal space.
As a result of this theorem, perfectly normal spaces belong to a special class of spaces called P-spaces. K. Morita defined the notion of P-space and he proved that a space is a Normal P-space if and only if is normal for every metric space (see the section below on P-spaces). Thus any perfectly normal space is a Normal P-space.
All spaces under consideration are Hausdorff. A subset of the space is a -subset of the space if is the intersection of countably many open subsets of . A subset of the space is an -subset of the space if is the union of countably many closed subsets of . Clearly, a set is a -subset of the space if and only if is an -subset of the space .
A space is said to be a perfectly normal space if is normal with the additional property that every closed subset of is a -subset of (or equivalently every open subset of is an -subset of ).
The perfect normality has a characterization in terms of zero-sets and cozero-sets. A subset of the space is said to be a zero-set if there exists a continuous function such that , where . A subset of the space is a cozero-set if is a zero-set, or more explicitly if there is a continuous function such that .
It is well known that the space is perfectly normal if and only if every closed subset of is a zero-set, equivalently every open subset of is a cozero-set. See here for a proof of this result. We use this result to show that is perfectly normal.
Let be a perfectly normal space and be a metric space. Since is a metric space, let be a base for such that each is locally finite. We show that is perfectly normal. To that end, we show that every open subset of is a cozero-set. Let be an open subset of .
For each , there exists open and there exists such that . Then is the union of all sets . Observe that for some integer . For each such that for some , let be the union of all corresponding open sets for all applicable .
For each positive integer , let be the collection of all open sets such that and for some . Let . As a result, .
Since both and are perfectly normal, for each , there exist continuous functions
Now define by the following:
for all . Note that the function is well defined. Since is locally finite in , is locally finite in . Thus is obtained by summing a finite number of values of . On the other hand, it can be shown that is continuous for each . Based on the definition of , it can be readily verified that for all and for all .
Define by the following:
It is clear that is continuous. We claim that . Recall that the open set is the union of all for all . Thus if for some , then since . If for all , since for all . Thus the open set is an -subset of . This concludes the proof that is perfectly normal.
The main theorem here is a classic result in general topology. An alternative proof is to show that any perfectly normal space is a P-space (definition given below). Then by Morita’s theorem, the product of any perfectly normal space and any metric space is normal (Theorem 1 below). For another proof that is elementary, see Lemma 7 in this previous post.
The notions of perfectly normal spaces and paracompact spaces are quite different. By the theorem discussed here, perfectly normal spaces are normally productive with metric spaces. It is possible for a paracompact space to have a non-normal product with a metric space. The classic example is the Michael line (discussed here).
On the other hand, there are perfectly normal spaces that are not paracompact. One example is Bing’s Example H, which is perfectly normal and not paracompact (see here).
Even though a perfectly normal space is normally productive with metric spaces, it cannot be normally productive in general. For each non-discrete perfectly normal space , there exists a normal space such that is not normal. This follows from Morita’s first conjecture (now a true statement). Morita’s first conjecture is discussed here.
P-Space in the Sense of Morita
Morita defined the notion of P-spaces  and . Let be a cardinal number such that . Let be the set of all finite ordered sequences where and all . Let be a space. The collection is said to be decreasing if this condition holds: and with imply that . The space is a P-space if for any cardinal and for any decreasing collection of closed subsets of , there exists open set for each such that the following conditions hold:
- for all , ,
- for any infinite sequence where each each finite subsequence is an element of , if , then .
If where . Then the index set defined above can be viewed as the set of all positive integers. As a result, the definition of P-space with implies the a condition in Dowker’s theorem (see condition 6 in Theorem 1 here). Thus any space that is normal and a P-space is countably paracompact (or countably shrinking or that is normal for every compact metric space or any other equivalent condition in Dowker’s theorem). The following is a theorem of Morita.
Theorem 1 (Morita)
Let be a space. Then is a normal P-space if and only if is normal for every metric space .
In light of Theorem 1, both perfectly normal spaces and normal countably compact spaces are P-spaces (see here). According to Theorem 1 and Dowker’s theorem, it follows that any normal P-space is countably paracompact.
- Morita K., On the Product of a Normal Space with a Metric Space, Proc. Japan Acad., Vol. 39, 148-150, 1963. (article information; paper)
- Morita K., Products of Normal Spaces with Metric Spaces, Math. Ann., Vol. 154, 365-382, 1964.