Michael line and Morita’s conjectures

This post discusses Michael line from the point of view of the three conjectures of Kiiti Morita.

K. Morita defined the notion of P-spaces in [7]. The definition of P-spaces is discussed here in considerable details. K. Morita also proved that a space X is a normal P-space if and only if the product X \times Y is normal for every metrizable space Y. As a result of this characterization, the notion of normal P-space (a space that is a normal space and a P-space) is useful in the study of products of normal spaces. Just to be clear, we say a space is a non-normal P-space (i.e. a space that is not a normal P-space) if the space is a normal space that is not a P-space.

K. Morita formulated his three conjectures in 1976. The statements of the conjectures are given below. Here is a basic discussion of the three conjectures. The notion of normal P-spaces is a theme that runs through the three conjectures. The conjectures are actually theorems since 2001 [2].

Here’s where Michael line comes into the discussion. Based on the characterization of normal P-spaces mentioned above, to find a normal space that is not a P-space (a non-normal P-space), we would need to find a non-normal product X \times Y such that one of the factors is a metric space and the other factor is a normal space. The first such example in ZFC is from an article by E. Michael in 1963 (found here and here). In this example, the normal space is M, which came be known as the Michael line, and the metric space is \mathbb{P}, the space of irrational numbers (as a subspace of the real line). Their product M \times \mathbb{P} is not normal. A basic discussion of the Michael line is found here.

Because M \times \mathbb{P} is not normal, the Michael line M is not a normal P-space. Prior to E. Michael’s 1963 article, we have to reach back to 1955 to find an example of a non-normal product where one factor is a metric space. In 1955, M. E. Rudin used a Souslin line to construct a Dowker space, which is a normal space whose product with the closed unit interval is not normal. The existence of a Souslin line was shown to be independent of ZFC in the late 1960s. In 1971, Rudin constructed a Dowker space in ZFC. Thus finding a normal space that is not a normal P-space (finding a non-normal product X \times Y where one factor is a metric space and the other factor is a normal space) is not a trivial matter.

Morita’s Three Conjectures

We show that the Michael line illustrates perfectly the three conjectures of K. Morita. Here’s the statements.

Morita’s Conjecture I. Let X be a space. If the product X \times Y is normal for every normal space Y then X is a discrete space.

Morita’s Conjecture II. Let X be a space. If the product X \times Y is normal for every normal P-space Y then X is a metrizable space.

Morita’s Conjecture III. Let X be a space. If the product X \times Y is normal for every normal countably paracompact space Y then X is a metrizable \sigma-locally compact space.

The contrapositive statement of Morita’s conjecture I is that for any non-discrete space X, there exists a normal space Y such that X \times Y is not normal. Thus any non-discrete space is paired with a normal space for forming a non-normal product. The Michael line M is paired with the space of irrational numbers \mathbb{P}. Obviously, the space \mathbb{P} is paired with the Michael line M.

The contrapositive statement of Morita’s conjecture II is that for any non-metrizable space X, there exists a normal P-space Y such that X \times Y is not normal. The pairing is more specific than for conjecture I. Any non-metrizable space is paired with a normal P-space to form a non-normal product. As illustration, the Michael line M is not metrizable. The space \mathbb{P} of irrational numbers is a metric space and hence a normal P-space. Here, M is paired with \mathbb{P} to form a non-normal product.

The contrapositive statement of Morita’s conjecture III is that for any space X that is not both metrizable and \sigma-locally compact, there exists a normal countably paracompact space Y such that X \times Y is not normal. Note that the space \mathbb{P} is not \sigma-locally compact (see Theorem 4 here). The Michael line M is paracompact and hence normal and countably paracompact. Thus the metric non-\sigma-locally compact \mathbb{P} is paired with normal countably paracompact M to form a non-normal product. Here, the metric space \mathbb{P} is paired with the non-normal P-space M.

In each conjecture, each space in a certain class of spaces is paired with one space in another class to form a non-normal product. For Morita’s conjecture I, each non-discrete space is paired with a normal space. For conjecture II, each non-metrizable space is paired with a normal P-space. For conjecture III, each metrizable but non-\sigma-locally compact is paired with a normal countably paracompact space to form a non-normal product. Note that the paired normal countably paracompact space would be a non-normal P-space.

Michael line as an example of a non-normal P-space is a great tool to help us walk through the three conjectures of Morita. Are there other examples of non-normal P-spaces? Dowker spaces mentioned above (normal spaces whose products with the closed unit interval are not normal) are non-normal P-spaces. Note that conjecture II guarantees a normal P-space to match every non-metric space for forming a non-normal product. Conjecture III guarantees a non-normal P-space to match every metrizable non-\sigma-locally compact space for forming a non-normal product. Based on the conjectures, examples of normal P-spaces and non-normal P-spaces, though may be hard to find, are guaranteed to exist.

We give more examples below to further illustrate the pairings for conjecture II and conjecture III. As indicated above, non-normal P-spaces are hard to come by. Some of the examples below are constructed using additional axioms beyond ZFC. The additional examples still give an impression that the availability of non-normal P-spaces, though guaranteed to exist, is limited.

Examples of Normal P-Spaces

One example is based on this classic theorem: for any normal space X, X is paracompact if and only if the product X \times \beta X is normal. Here \beta X is the Stone-Cech compactification of the completely regular space X. Thus any normal but not paracompact space X (a non-metrizable space) is paired with \beta X, a normal P-space, to form a non-normal product.

Naturally, the next class of non-metrizable spaces to be discussed should be the paracompact spaces that are not metrizable. If there is a readily available theorem to provide a normal P-space for each non-metrizable paracompact space, then there would be a simple proof of Morita’s conjecture II. The eventual solution of conjecture II is far from simple [2]. We narrow the focus to the non-metrizable compact spaces.

Consider this well known result: for any infinite compact space X, the product \omega_1 \times X is normal if and only if the space X has countable tightness (see Theorem 1 here). Thus any compact space with uncountable tightness is paired with \omega_1, the space of all countable ordinals, to form a non-normal product. The space \omega_1, being a countably compact space, is a normal P-space. A proof that normal countably compact space is a normal P-space is given here.

We now handle the case for non-metrizable compact spaces with countable tightness. In this case, compactness is not needed. For spaces with countable tightness, consider this result: every space with countable tightness, whose products with all perfectly normal spaces are normal, must be metrizable [3] (see Corollary 7). Thus any non-metrizable space with countable tightness is paired with some perfectly normal space to form a non-normal product. Any reader interested in what these perfectly normal spaces are can consult [3]. Note that perfectly normal spaces are normal P-spaces (see here for a proof).

Examples of Non-Normal P-Spaces

Another non-normal product is X_B \times B where B \subset \mathbb{R} is a Bernstein set and X_B is the space with the real line as the underlying set such that points in B are isolated and points in \mathbb{R}-B retain the usual open sets. The set B \subset \mathbb{R} is said to be a Bernstein set if every uncountable closed subset of the real line contains a point in B and contains a point in the complement of B. Such a set can be constructed using transfinite induction as shown here. The product X_B \times B is not normal where B is considered a subspace of the real line. The proof is essentially the same proof that shows M \times \mathbb{P} is not normal (see here). The space X_B is a Lindelof space. It is not a normal P-space since its product with B, a separable metric space, is not normal. However, this example is essentially the same example as the Michael line since the same technique and proof are used. On the one hand, the X_B \times B example seems like an improvement over Michael line example since the first factor X_B is Lindelof. On the other hand, it is inferior than the Michael line example since the second factor B is not completely metrizable.

Moving away from the idea of Michael, there exist a Lindelof space and a completely metrizable (but not separable) space whose product is of weight \omega_1 and is not normal [5]. This would be a Lindelof space that is a non-normal P-space. However, this example is not as elementary as the Michael line, making it not as effective as an illustration of Morita’s three conjectures.

The next set of non-normal P-spaces requires set theory. A Michael space is a Lindelof space whose product with \mathbb{P}, the space of irrational numbers, is not normal. Michael problem is the question: is there a Michael space in ZFC? It is known that a Michael space can be constructed using continuum hypothesis [6] or using Martin’s axiom [1]. The construction using continuum hypothesis has been discussed in this blog (see here). The question of whether there exists a Michael space in ZFC is still unsolved.

The existence of a Michael space is equivalent to the existence of a Lindelof space and a separable completely metrizable space whose product is non-normal [4]. A Michael space, in the context of the discussion in this post, is a non-normal P-space.

The discussion in this post shows that the example of the Michael line and other examples of non-normal P-spaces are useful tools to illustrate Morita’s three conjectures.

Reference

  1. Alster K.,On the product of a Lindelof space and the space of irrationals under Martin’s Axiom, Proc. Amer. Math. Soc., Vol. 110, 543-547, 1990.
  2. Balogh Z.,Normality of product spaces and Morita’s conjectures, Topology Appl., Vol. 115, 333-341, 2001.
  3. Chiba K., Przymusinski T., Rudin M. E.Nonshrinking open covers and K. Morita’s duality conjectures, Topology Appl., Vol. 22, 19-32, 1986.
  4. Lawrence L. B., The influence of a small cardinal on the product of a Lindelof space and the irrationals, Proc. Amer. Math. Soc., 110, 535-542, 1990.
  5. Lawrence L. B., A ZFC Example (of Minimum Weight) of a Lindelof Space and a Completely Metrizable Space with a Nonnormal Product, Proc. Amer. Math. Soc., 124, No 2, 627-632, 1996.
  6. Michael E., Paracompactness and the Lindelof property in nite and countable cartesian products, Compositio Math., 23, 199-214, 1971.
  7. Morita K., Products of Normal Spaces with Metric Spaces, Math. Ann., Vol. 154, 365-382, 1964.
  8. Rudin M. E., A Normal Space X for which X \times I is not Normal, Fund. Math., 73, 179-186, 1971.

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