Assuming that the function space is normal, what can be said about the domain space ? In this post, we prove a theorem that yields a corollary that for any normal space , if is normal, then has countable extent (i.e. every closed and discrete subset of is countable). Thus the normality of the function space limits the size of a closed and discrete subset of the domain space. It then follows that for any metric space , if is normal, has is second countable (i.e. having a countable base). Another immediate, but slightly less obvious, corollary is that for any that is a normal Moore space, if is normal, then is metrizable.
For definitions of basic open sets and other background information on the function space , see this previous post.
Let be a space. Let . Let be the natural projection from the product space into the product space . Specifically, if , then , i.e., the function restricted to . In the discussion below, is defined just on , i.e., is the natural projection from into . It is always the case that . It is not necessarily the case that . However, if is a normal space and is closed in , then and is the natural projection from onto . We prove the following theorem.
Suppose that is a normal space. Let be a closed subspace of . Then is a normal space.
Theorem 1 is found in  (see Theorem I.6.2). In proving Theorem 1, we need the following lemma.
Let be a product of separable metrizable spaces. Let be a dense subspace of . Then the following conditions are equivalent.
- is normal.
- For any pair of disjoint closed subsets and of , there exists a countable such that .
- For any pair of disjoint closed subsets and of , there exists a countable such that and are separated in , meaning that .
For a proof of Lemma 2, see Lemma 1 in this previous post.
Proof of Theorem 1
Note that is a dense subspace of . Let and be disjoint closed subsets of . To show is normal, by Lemma 2, we only need to produce a countable such that . The closure here is taken in .
Let and . Both and are closed subsets of . By Lemma 2, there exists some countable such that . The closure here is taken in . According to the remark at the end of this previous post, for any countable such that , . In other words, the countable set can be enlarged and the conclusion of the lemma still holds. With this observation in mind, we can assume that . If not, we can always throw countably many points of into and still have .
Let . We claim that . The closure here is taken . Suppose that . Choose such that . It follows that . To see this, let where is a standard basic open set. Let be the support of , i.e., the finite set of such that . Let and . Let . Note that . Since , there is some such that . Note that is the support of .
Because the space is completely regular, there is a such that for all and for all . Let . Since , . Note that on , hence on and that on . Thus . Since is an arbitrary open set containing , it follows that . By a similar argument, it can be shown that . This is a contradiction since . Therefore the claim that is true, with the closure being taken in .
Because , observe that and . Furthermore, . Thus we can claim that , with the closure being taken in . By Lemma 2, is normal.
Let be a normal space. If is normal, then has countable extent, i.e., every closed and discrete subset of is countable.
Proof of Corollary 3
Let be a closed and discrete subset of . We show that must be countable. Since is closed and is normal, . By Theorem 1, is normal. Since is discrete, . If is uncountable, is not normal. Thus must be countable.
Let be a metrizable space. If is normal, then has a countable base.
Proof of Corollary 4
Note that in any metrizable space, the weight equals the extent. By Corollary 3, has countable extent and thus has countable base.
Let be a normal space. If is normal, then is collectionwise normal.
Proof of Corollary 5
Any normal space with countable extent is collectionwise normal. See Theorem 2 in this previous post.
Let be a normal Moore space. If is normal, then is metrizable.
Proof of Corollary 6
Suppose is normal. By Theorem 1, has countable extent. By Corollary 5, is collectionwise normal. According to Bing’s metrization theorem, any collectionwise normal Moore space is metrizable (see  Theorem 5.4.1 in page 329).
- 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.