Is every normal dense subspace of a product of separable metric spaces collectionwise normal? This question was posed by Arkhangelskii in [1] (see Problem I.5.25). A partial positive answer is provided by a theorem that is usually attributed to Corson: If is a normal dense subspace of a product of separable metric spaces and if is also normal, then is collectionwise normal. In this post, using a simple combinatorial argument, we show that any normal dense subspace of a product of continuum many separable metric space is collectionwise normal (see Corollary 4 below), which is a corollary of the following theorem.

**Theorem 1**

Let be a normal space with character . If , then the following holds:

- If is a closed and discrete subspace of with , then contains a separated subset of cardinality .

Theorem 1 gives the corollary indicated at the beginning and several other interesting results. The statement means that the cardinality of the power set (the set of all subsets) of is strictly less than the cardinality of the power set of . Note that the statement follows from the continuum hypothesis (CH), the statement that . With the assumption , Theorem 1 is a theorem that goes beyond ZFC. We also present an alternative to Theorem 1 that removes the assumption (see Theorem 6 below).

A subset of a space is a separated set (in ) if for each , there is an open subset of with such that is a pairwise disjoint collection. First we prove Theorem 1 and then discuss the corollaries.

____________________________________________________________________

**Proof of Theorem 1**

Suppose is a closed and discrete subset of with such that no subset of of cardinality can be separated. We then show that .

For each , let be a local base at the point such that . Let . Thus . By normality, for each , let be an open subset of such that and . For each , consider the following collection of open sets:

For each , choose a maximal disjoint collection of open sets in . Because no subset of of cardinality can be separated, each is countable. If , then .

Let be the power set (i.e. the set of all subsets) of . Let be the set of all countable subsets of . Then the mapping is a one-to-one map from into . Note that . Also note that since , . Thus .

____________________________________________________________________

**Some Corollaries of Theorem 1**

Here’s some corollaries that follow easily from Theorem 1. A space has the countable chain condition (CCC) if every pairwise disjoint collection of non-empty open subset of is countable. For convenience, if has the CCC, we say is CCC. The following corollaries make use of the fact that any normal space with countable extent is collectionwise normal (see Theorem 2 in this previous post).

**Corollary 2**

Let be a CCC space with character . If , then the following conditions hold:

- If is normal, then every closed and discrete subset of is countable, i.e., has countable extent.
- If is normal, then is collectionwise normal.

**Corollary 3**

Let be a CCC space with character . If **CH** holds, then the following conditions hold:

- If is normal, then every closed and discrete subset of is countable, i.e., has countable extent.
- If is normal, then is collectionwise normal.

**Corollary 4**

Let be a product where each factor is a separable metric space. If , then the following conditions hold:

- If is a normal dense subspace of , then has countable extent.
- If is a normal dense subspace of , then is collectionwise normal.

Corollary 4 is the result indicated in the title of the post. The product of separable spaces has the CCC. Thus the product space and any dense subspace of have the CCC. Because is a product of continuum many separable metric spaces, and any subspace of have characters . Then Corollary 4 follows from Corollary 2.

When dealing with the topic of normal versus collectionwise normal, it is hard to avoid the connection with the normal Moore space conjecture. Theorem 1 gives the result of F. B. Jones from 1937 (see [3]). We have the following theorem.

**Theorem 5**

If , then every separable normal Moore space is metrizable.

Though this was not how Jones proved it in [3], Theorem 5 is a corollary of Corollary 2. By Corollary 2, any separable normal Moore space is collectionwise normal. It is well known that collectionwise normal Moore space is metrizable (Bing’s metrization theorem, see Theorem 5.4.1 in [2]).

____________________________________________________________________

**A ZFC Theorem**

We now prove a result that is similar to Corollary 2 but uses no set-theory beyond the Zermeloâ€“Fraenkel set theory plus axiom of choice (abbreviated by ZFC). Of course the conclusion is not as strong. Even though the assumption is removed in Theorem 6, note the similarity between the proof of Theorem 1 and the proof of Theorem 6.

**Theorem 6**

Let be a CCC space with character . Then the following conditions hold:

- If is normal, then every closed and discrete subset of has cardinality less than continuum.

**Proof of Theorem 6**

Let be a normal CCC space with character . Let be a closed and discrete subset of . We show that . Suppose that .

For each , let be a local base at the point such that . Let . Thus . By normality, for each , let be an open subset of such that and . For each , consider the following collection of open sets:

For each , choose such that is a maximal disjoint collection. Since is CCC, is countable. It is clear that if , then .

Let be the power set (i.e. the set of all subsets) of . Let be the set of all countable subsets of . Then the mapping is a one-to-one map from into . Note that since , . Thus . However, is assumed to be of cardinality continuum. Then , leading to a contradiction. Thus it must be the case that .

With Theorem 6, Corollary 3 still holds. Theorem 6 removes the set-theoretic assumption of . As a result, the upper bound for cardinalities of closed and discrete sets is (at least potentially) higher.

____________________________________________________________________

**Reference**

- Arkhangelskii, A. V.,
*Topological Function Spaces*, Mathematics and Its Applications Series, Kluwer Academic Publishers, Dordrecht, 1992. - Engelking, R.,
*General Topology, Revised and Completed edition*, Heldermann Verlag, Berlin, 1989. - Jones, F. B.,
*Concerning normal and completely normal spaces*, Bull. Amer. Math. Soc., 43, 671-677, 1937.

____________________________________________________________________