This post complements two results discussed in two previous blog posts concerning -diagonal. One result is that any compact space with a -diagonal is metrizable (see here). The other result is that the compactness in the first result can be relaxed to countably compactness. Thus any countably compact space with a -diagonal is metrizable (see here). The countably compactness in the second result cannot be relaxed to pseudocompactness. The Mrowka space is a pseudocompact space with a -diagonal that is not submetrizable, hence not metrizable (see here). However, if we strengthen the -diagonal to a regular -diagonal while keeping pseudocompactness fixed, then we have a theorem. We prove the following theorem.

*Theorem 1*

If the space is pseudocompact and has a regular -diagonal, then is metrizable.

All spaces are assumed to be Hausdorff and completely regular. The assumption of completely regular is crucial. The proof of Theorem 1 relies on two lemmas concerning pseudocompact spaces (one proved in a previous post and one proved here). These two lemmas work only for completely regular spaces.

The proof of Theorem 1 uses a metrization theorem. The best metrization to use in this case is Moore metrization theorem (stated below). The result in Theorem 1 is found in [2].

First some basics. Let be a space. The diagonal of the space is the set . When the diagonal , as a subset of , is a -set, i.e. is the intersection of countably many open subsets of , the space is said to have a -diagonal.

The space is said to have a regular -diagonal if the diagonal is a regular -set in , i.e. where each is an open subset of with . If , then . Thus if a space has a regular -diagonal, it has a -diagonal. We will see that there exists a space with a -diagonal that fails to be a regular -diagonal.

The space is a pseudocompact space if for every continuous function , the image is a bounded set in the real line . Pseudocompact spaces are discussed in considerable details in this previous post. We will rely on results from this previous post to prove Theorem 1.

The following lemma is used in proving Theorem 1.

*Lemma 2*

Let be a pseudocompact space. Suppose that is a decreasing sequence of non-empty open subsets of such that for some point . Then is a local base at the point .

*Proof of Lemma 2*

Let be a decreasing sequence of open subsets of such that . Let be open in with . If for some , then we are done. Suppose that for each .

Choose open with . Consider the sequence . This is a decreasing sequence of non-empty open subsets of . By Theorem 2 in this previous post, . Let be a point in this non-empty set. Note that . This means that . Since for each , any open set containing would contain a point not in . This is a contradiction since . Thus it must be the case that for some .

The following metrization theorem is useful in proving Theorem 1.

** Theorem 3** (Moore Metrization Theorem)

Let be a space. Then is metrizable if and only if the following condition holds.

There exists a decreasing sequence of open covers of such that for each , the sequence is a local base at the point .

For any family of subsets of , and for any , the notation refers to the set . In other words, it is the union of all sets in that contain points of . The set is also called the star of the set with respect to the family . If , we write instead of . The set indicated in Theorem 3 is the star of the set with respect to the open cover .

Theorem 3 follows from Theorem 1.4 in [1], which states that for any -space , is metrizable if and only if there exists a sequence of open covers of such that for each open and for each , there exist an open and an integer such that and .

**Proof of Theorem 1**

Suppose is pseudocompact such that its diagonal where each is an open subset of with . We can assume that . For each , define the following:

Note that each is an open cover of . Also note that is a decreasing sequence since is a decreasing sequence of open sets. We show that is a sequence of open covers of that satisfies Theorem 3. We establish this by proving the following claims.

** Claim 1**. For each , .

To prove the claim, let . There is an integer such that . Choose open sets and such that and . Note that and .

We want to show that , which implies that . Suppose . This means that for some with . Then . Note that . This implies that , a contradiction. Thus . Since , . We have established that for each , .

** Claim 2**. For each , is a local base at the point .

Note that is a decreasing sequence of open sets such that . By Lemma 2, is a local base at the point .

** Claim 3**. For each , .

Let . There is an integer such that . Choose open sets and such that and . It follows that for all . Furthermore, for all . By Claim 2, choose integers and such that and . Choose an integer . It follows that . Since and , it follows that .

We claim that . Suppose not. Choose . It follows that for some such that and . Furthermore . Note that . This means that , contradicting the fact observed in the preceding paragraph. It must be the case that .

Because there is an open set containing , namely , that contains no points of , . Thus Claim 3 is established.

** Claim 4**. For each , is a local base at the point .

Note that is a decreasing sequence of open sets such that . By Lemma 2, is a local base at the point .

In conclusion, the sequence of open covers satisfies the properties in Theorem 3. Thus any pseudocompact space with a regular -diagonal is metrizable.

**Example**

Any submetrizable space has a -diagonal. The converse is not true. A classic example of a non-submetrizable space with a -diagonal is the Mrowka space (discussed here). The Mrowka space is also called the psi-space since it is sometimes denoted by where is a maximal family of almost disjoint subsets of . Actually would be a family of spaces since is any maximal almost disjoint family. For any maximal , is a pseudocompact non-submetrizable space that has a -diagonal. This example shows that the requirement of a regular -diagonal in Theorem 1 cannot be weakened to a -diagonal. See here for a more detailed discussion of this example.

**Reference**

- Gruenhage, G.,
*Generalized Metric Spaces*, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, 423-501, 1984. - McArthur W. G.,
*-Diagonals and Metrization Theorems*, Pacific Journal of Mathematics, Vol. 44, No. 2, 613-317, 1973.

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