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.
Theorem 1

Suppose that is a normal space. Let be a closed subspace of . Then is a normal space.
Theorem 1 is found in [1] (see Theorem I.6.2). In proving Theorem 1, we need the following lemma.
Lemma 2

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.
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Some Corollaries
Corollary 3

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.
Corollary 4

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.
Corollary 5

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.
Corollary 6

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 [2] Theorem 5.4.1 in page 329).
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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.
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