This post discusses the three conjectures that were proposed by K. Morita in 1976. These conjectures concern normality in product spaces. To start the discussion, here’s the conjectures.
These statements are no longer conjectures. Partial results appeared after the conjectures were proposed in 1976. The complete resolution of the conjectures came in 2001 in a paper by Zoli Balogh . Though it is more appropriate to call these statements theorems, it is still convenient to call them conjectures. Just know that they are now known results rather open problems to be solved. The focus here is not on the evolution of the solutions. Instead, we discuss the relations among the three conjectures and why they are amazing results in the study of normality in product spaces.
As discussed below, in each of these conjectures, one direction is true based on prior known theorems (see Theorem 1, Theorem 2 and Theorem 4 below). The conjectures can be stated as follows.
P-spaces are defined by K. Morita . He proved that a space is a normal P-space if and only if the product is normal for every metrizable space (see theorem 2 below). Normal P-spaces are also discussed here. A space is -locally compact space if is the union of countably many locally compact subspaces each of which is also closed subspace of .
As we will see below, these conjectures are also called duality conjectures because they are duals of known results.
 is a survey of Morita’s conjecture.
Here’s three theorems that are duals to the conjectures.
Let be a space. The product space is normal for every discrete space if and only if is normal.
Let be a space. The product space is normal for every metrizable space if and only if is a normal P-space.
Let be a space. The product space is normal for every metrizable -locally compact space if and only if is normal countably paracompact.
The key words in red are for emphasis. In each of these three theorems, if we switch the two key words in red, we would obtain the statements for the conjectures. In this sense, the conjectures are called duality conjectures since they are duals of known results.
Theorem 1 is actually not found in the literature. It is an easy theorem. Theorem 2, found in , is a characterization of normal P-space (discussed here). Theorem 3 is a well known result based on the following theorem by K. Morita .
Let be a metrizable space. Then the product is normal for every normal countably paracompact space if and only if is a -locally compact space.
We now show that Theorem 3 can be established using Theorem 4. Theorem 4 is also Theorem 3.5 in p. 111 of . A proof of Theorem 4 is found in Theorem 1.8 in p. 130 of .
Proof of Theorem 3
Suppose is normal and countably paracompact. Let be a metrizable -locally compact space. By Theorem 4, is normal.
This direction uses Dowker’s theorem. We give a contrapositive proof. Suppose that is not both normal and countably paracompact. Case 1. is not normal. Then is not normal where is any one-point discrete space. Case 2. is normal and not countably paracompact. This means that is a Dowker space. Then is not normal. In either case, is not normal for some compact metric space. Thus is not normal for some -locally compact metric space. This completes the proof of Theorem 3.
The First and Third Conjectures
The first conjecture of Morita was proved by Atsuji  and Rudin  in 1978. The proof in  is a constructive proof. The key to that solution is to define a -Dowker space. Suppose is a non-discrete space. Let be the least cardinal of a non-discrete subspace of . Then construct a -Dowker space as in . It follows that is not normal. The proof that is not normal is discussed here.
Conjecture III was confirmed by Balogh in 1998 . We show here that the first and third conjectures of Morita can be confirmed by assuming the second conjecture.
Conjecture II implies Conjecture I
We give a contrapositive proof of Conjecture I. Suppose that is not discrete. We wish to find a normal space such that is not normal. Consider two cases for . Case 1. is not metrizable. By Conjecture II, is not normal for some normal P-space . Case 2. is metrizable. Since is infinite and metric, would contain an infinite compact metric space . For example, contains a non-trivial convergent sequence and let be a convergence sequence plus the limit point. Let be a Dowker space. Then the product is not normal. It follows that is not normal. Thus there exists a normal space such that is not normal in either case.
Conjecture II implies Conjecture III
Suppose that the product is normal for every normal and countably paracompact space . Since any normal P-space is a normal countably paracompact space, is normal for every normal and P-space . By Conjecture II, is metrizable. By Theorem 4, is -locally compact.
The Second Conjecture
The above discussion shows that a complete solution to the three conjectures hinges on the resolution of the second conjecture. A partial resolution came in 1986 . In that paper, it was shown that under V = L, conjecture II is true.
The complete solution of the second conjecture is given in a paper of Balogh  in 2001. The path to Balogh’s proof is through a conjecture of M. E. Rudin identified as Conjecture 9.
Conjecture 9 was part of a set of 14 conjectures stated in . It is also discussed in . In , conjecture 9 was shown to be equivalent to Morita’s second conjecture. In , Balogh used his technique for constructing a Dowker space of cardinality continuum to obtain a space as described in conjecture 9.
The resolution of conjecture II is considered to be one of Balogh greatest hits .
Abundance of Non-Normal Products
One immediate observation from Morita’s conjecture I is that existence of non-normal products is wide spread. Conjecture I indicates that every normal non-discrete space is paired with some normal space such that their product is not normal. So every normal non-discrete space forms a non-normal product with some normal space. Given any normal non-discrete space (no matter how nice it is or how exotic it is), it can always be paired with another normal space (sometimes paired with itself) for a non-normal product.
Suppose we narrow the focus to spaces that are normal and non-metrizable. Then any such space is paired with some normal P-space to form a non-normal product space (Morita’s conjecture II). By narrowing the focus on to the non-metrizable spaces, we obtain more clarity on the paired space to form non-normal product, namely a normal P-space. As an example, let be the Michael line (normal and non-metrizable). It is well known that in this case is paired with , the space of irrational numbers with the usual Euclidean topology, to form a non-normal product (discussed here).
Another example is being the Sorgenfrey line. It is well known that in this case is paired with itself to form a non-normal product (discussed here). Morita’s conjectures are powerful indication that these two non-normal products are not isolated phenomena.
Another interesting observation about conjecture II is that normal P-spaces are not productive with respect to normality. More specifically, for any non-metrizable normal P-space , conjecture II tells us that there exists another normal P-space such that is not normal.
Now we narrow the focus to spaces that are metrizable but not -locally compact. For any such space , conjecture III tells us that is paired with a normal countably paracompact space to form a non-normal product. Using the Michael line example, this time let , the space of irrational numbers, which is a metric space that is not -locally compact. The paired normal and countably paracompact space is the Michael line.
Each conjecture is about existence of a normal that is paired with a given to form a non-normal product. For Conjecture I, the given is from a wide class (normal non-discrete). As a result, there is not much specific information on the paired , other than that it is normal. For Conjectures II and III, the given space is from narrower classes. As a result, there is more information on the paired .
The concept of Dowker spaces runs through the three conjectures, especially the first conjecture. Dowker spaces and -Dowker spaces provide reliable pairing for non-normal products. In fact this is one way to prove conjecture I , also see here. For any normal space with a countable non-discrete subspace, the product of and any Dowker space is not normal (discussed here). For any normal space such that the least cardinality of a non-discrete subspace is an uncountable cardinal , the product is not normal where is a -Dowker space as constructed in , also discussed here.
In finding a normal pair for a normal space , if we do not care about having a high degree of normal productiveness (e.g. normal P or normal countably paracompact), we can always let be a Dowker space or -Dowker space. In fact, if the starting space is a metric space, the normal pair for a non-normal product (by definition) has to be a Dowker space. For example, if , then the normal space such that is by definition a Dowker space. The search for a Dowker space spanned a period of 20 years. For the real line , the normal pair for a non-normal product is also a Dowker space. For “nice” spaces such as metric spaces, finding a normal space to form non-normal product is no trivial problem.
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Dan Ma math
Daniel Ma mathematics