The property of being submetrizable implies having a -diagonal. There are several other properties lying between these two properties (see ). Before diving into these other properties, it may be helpful to investigate a classic example of a space with a -diagonal that is not submetrizable.
The diagonal of a space is the set , a subset of the square . An interesting property is when the diagonal of a space is a -set in (the space is said to have a -diagonal). Any compact space or a countably compact space with this property must be metrizable (see compact and countably compact space). A space is said to be submetrizable if there is a topology that can be defined on such that is a metrizable space and . In other words, a submetrizable space is a space that has a coarser (weaker) metrizable topology. Every submetrizable space has a -diagonal. Note that when has a weaker metric topology, the diagonal is always a -set in the metric square (hence in the square in the original topology). The property of having a -diagonal is strictly weaker than the property of having a weaker metric topology. In this post, we discuss the Mrowka space, which is a classic example of a space with a -diagonal that is not submetrizable.
The Mrowka space (also called Psi space) was discussed previously in this blog (see this post). For the sake of completeness, the example is defined here.
First, we define some basic notions. Let be the first infinite ordinal (or more conveniently the set of all nonnegative integers). Let be a family of infinite subsets of . The family is said to be an almost disjoint family if for each two distinct , is finite. An almost disjoint family is said to be a maximal almost disjoint family if is an infinite subset of such that , then is infinite for some . In other words, if you put one more set into a maximal almost disjoint family, it ceases to be almost disjoint.
A natural question is whether there is an uncountable almost disjoint family of subsets of . In fact, there is one whose cardinality is continuum (the cardinality of the real line). To see this, identify with (the set of all rational numbers). Let be the set of all irrational numbers. For each , choose a subsequence of consisting of distinct elements that converges to (in the Euclidean topology). Then the family of all such sequences of rational numbers would be an almost disjoint family. By a Zorn’s Lemma argument, this almost disjoint family is contained within a maximal almost disjoint family. Thus we also have a maximal almost disjoint family of cardinality continuum. On the other hand, there is no countably infinite maximal almost disjoint family of subsets of (see this post).
Let be an infinite almost disjoint family of subsets of . We now define a Mrowka space (or -space), denoted by . The underlying set is . Points in are isolated. For , a basic open set is of the form where is finite. It is straightforward to verify that is Hausdorff, first countable and locally compact. It has a countable dense set of isolated points. Note that is an infinite discrete and closed set in the space . Thus is not countably compact.
We would like to point out that the definition of a Mrowka space only requires that the family is an almost disjoint family and does not necessarily have to be maximal. For the example discribed in the title, needs to be a maximal almost disjoint family of subsets of .
Let be a maximal almost disjoint family of subsets of . Then as defined above is a space in which there is a -diagonal that is not submetrizable.
Note that is pseudocompact (proved in this post). Because there is no countable maximal almost disjoint family of subsets of , must be an uncountable in addition to being a closed and discrete subspace of (thus the space is not Lindelof). Since is separable and is not Lindelof, is not metrizable. Any psuedocompact submetrizable space is metrizable (see Theorem 4 in this post). Thus must not be submetrizable.
On the other hand, any -space (even if is not maximal) is a Moore space. It is well known that any Moore space has a -diagonal. The remainder of this post has a brief discussion of Moore space.
A sequence of open covers of a space is a development for if for each and each open set with , there is some such that any open set in containing the point is contained in . A developable space is one that has a development. A Moore space is a regular developable space.
Suppose that is a Moore space. We show that has a -diagonal. That is, we wish to show that is a -set in .
Let be a development. For each , let . Clearly . Let . For each , for some . We claim that . Suppose that . By the definition of development, there exists some such that every open set in containing the point has to be a subset of . Then , which contradicts . Thus we have .
The remaining thing to show is that is a Moore space. For each positive integer , let and let . The development is defined by , where for each , consists of open sets of the form where plus any singleton () that has not been covered by the sets .
- Arhangel’skii, A. V., Buzyakova, R. Z., The rank of the diagonal and submetrizability, Commentationes Mathematicae Universitatis Carolinae, Vol. 47 (2006), No. 4, 585-597.
- Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
- Willard, S., General Topology, Addison-Wesley Publishing Company, 1970.