Any collectionwise normal space is a normal space. Any perfectly normal space is a hereditarily normal space. In general these two implications are not reversible. In function spaces , the two implications are reversible. There is a normal space that is not countably paracompact (such a space is called a Dowker space). If a function space is normal, it is countably paracompact. Thus normality in is a strong property. This post draws on Dowker’s theorem and other results, some of them are previously discussed in this blog, to discuss this remarkable aspect of the function spaces .
Since we are discussing function spaces, the domain space has to have sufficient quantity of real-valued continuous functions, e.g. there should be enough continuous functions to separate the points from closed sets. The ideal setting is the class of completely regular spaces (also called Tychonoff spaces). See here for a discussion on completely regular spaces in relation to function spaces.
Let be a completely regular space. Let be the set of all continuous functions from into the real line . When is endowed with the pointwise convergence topology, the space is denoted by (see here for further comments on the definition of the pointwise convergence topology).
When Function Spaces are Normal
Let be a completely regular space. We discuss these four facts of :
- If the function space is normal, then is countably paracompact.
- If the function space is hereditarily normal, then is perfectly normal.
- If the function space is normal, then is collectionwise normal.
- Let be a normal space. If is normal, then has countable extent, i.e. every closed and discrete subset of is countable, implying that is collectionwise normal.
Fact #1 and Fact #2 rely on a representation of as a product space with one of the factors being the real line. For , let . Then . This representation is discussed here.
Another useful tool is Dowker’s theorem, which essentially states that for any normal space , the space is countably paracompact if and only if is normal for all compact metric space if and only if is normal. For the full statement of the theorem, see Theorem 1 in this previous post, which has links to the proofs and other discussion.
To show Fact #1, suppose that is normal. Immediately we make use of the representation where . Since is normal, is also normal. By Dowker’s theorem, is countably paracompact. Note that is a closed subspace of the normal . Thus is also normal.
One more helpful tool is Theorem 5 in in this previous post, which is like an extension of Dowker’s theorem, which states that a normal space is countably paracompact if and only if is normal for any -compact metric space . This means that is normal.
We want to show is countably paracompact. Since is normal (based on the argument in the preceding paragraph), is normal. Thus according to Dowker’s theorem, is countably paracompact.
For Fact #2, a helpful tool is Katetov’s theorem (stated and proved here), which states that for any hereditarily normal , one of the factors is perfectly normal or every countable subset of the other factor is closed (in that factor).
To show Fact #2, suppose that is hereditarily normal. With and according to Katetov’s theorem, must be perfectly normal. The product of a perfectly normal space and any metric space is perfectly normal (a proof is found here). Thus is perfectly normal.
The proof of Fact #3 is found in Problems 294 and 295 of . The key to the proof is a theorem by Reznichenko, which states that any dense convex normal subspace of has countable extent, hence is collectionwise normal (problem 294). See here for a proof that any normal space with countable extent is collectionwise normal (see Theorem 2). The function space is a dense convex subspace of (problem 295). Thus if is normal, then it has countable extent and hence collectionwise normal.
Fact #4 says that normality of the function space imposes countable extent on the domain. This result is discussed in this previous post (see Corollary 3 and Corollary 5).
The facts discussed here give a flavor of what function spaces are like when they are normal spaces. For further and deeper results, see  and .
Fact #1 is essentially driven by Dowker’s theorem. It follows from the theorem that whenever the product space is normal, one of the factor must be countably paracompact if the other factor has a non-trivial convergent sequence (see Theorem 2 in this previous post). As a result, there is no Dowker space that is a . No pathology can be found in with respect to finding a Dowker space. In fact, not only is normal for any compact metric space , it is also true that is normal for any -compact metric space when is normal.
The driving force behind Fact #2 is Katetov’s theorem, which basically says that the hereditarily normality of is a strong statement. Coupled with the fact that is of the form , Katetov’s theorem implies that is perfectly normal. The argument also uses the basic fact that perfectly normality is preserved when taking product with metric spaces.
There are examples of normal but not collectionwise normal spaces (e.g. Bing’s Example G). Resolution of the question of whether normal but not collectionwise normal Moore space exists took extensive research that spanned decades in the 20th century (the normal Moore space conjecture). The function is outside of the scope of the normal Moore space conjecture. The function space is usually not a Moore space. It can be a Moore space only if the domain is countable but then would be a metric space. However, it is still a powerful fact that if is normal, then it is collectionwise normal.
On the other hand, a more interesting point is on the normality of . Suppose that is a normal Moore space. If happens to be normal, then Fact #4 says that would have to be collectionwise normal, which means is metrizable. If the goal is to find a normal Moore space that is not collectionwise normal, the normality of would kill the possibility of being the example.
- Arkhangelskii, A. V., Topological Function Spaces, Mathematics and Its Applications Series, Kluwer Academic Publishers, Dordrecht, 1992.
- Tkachuk V. V., A -Theory Problem Book, Topological and Function Spaces, Springer, New York, 2011.