Compactness, countably compactness and pseudocompactness are three successively weaker properties. It follows easily from definitions that

None of these arrows can be reversed. It is well known that either compactness or countably complactness plus having a -diagonal implies metrizability. We have:

A question can be asked whether these results can be extended to pseudocompactness.

**Question**

The answer to this question is no. The space defined using a maximal almost disjoint family of subsets of is an example of a non-metrizable pseudocompact space with a -diagonal (discussed in this post). In this post we show that if we strengthen “having a -diagonal” to being submetrizable, we have a theorem. Specifically, we show:

For the result of , see this post. For the result of , see this post. In this post, we discuss the basic properties of pseudocompactness that build up to the result of . All spaces considered here are at least Tychonoff (i.e. completely regular). For any basic notions not defined here, see [1] or [2].

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**Pseudocompact Spaces**

A space is said to be pseudocompact if every real-valued continuous function defined on is a bounded function. Any real-valued continuous function defined on a compact space must be bounded (and is thus pseudocomppact). If there were an unbounded real-valued continuous function defined on a space , then would have a countably infinite discrete set (thus not countably compact). Thus countably compact implies pseudocompact, as indicated by .

A space is submetrizable if there is a coarser (i.e. weaker) topology that is a metrizable topology. Specifically the topological space is submetrizable if there is another topology that can be defined on such that and is metrizable. The Sorgenfrey line is non-metrizable and yet the Sorgenfrey topology has a weaker topology that is metrizable, namely the Euclidean topology of the real line.

The following two theorems characterizes pseudocompact spaces in terms of locally finite open family of open sets (Theorem 1) and the finite intersection property (Theorem 2). Both theorems are found in Engelking (Theorem 3.10.22 and Theorem 3.10.23 in page 207 of [1]). Theorem 3 states that in a pseudocompact space, closed domains are pseudocompact (the definition of closed domain is stated before the theorem). Theorem 4 is the main theorem (result stated above).

**Theorem 1**

Let be a space. The following conditions are equivalent:

- The space is pseudocompact.
- If is a locally finite family of non-empty open subsets of , then is finite.
- If is a locally finite open cover of , then is finite.
- If is a locally finite open cover of , then has a finite subcover.

**Proof**

Suppose that condition does not hold. Then there is an infinite locally finite family of non-empty open sets such that . We wish to define an unbounded continuous function using .

This is where we need to invoke the assumption of complete regularity. For each choose a point . Then for each , there is a continuous function such that and . Define by .

Because is locally finite, the function is essentially pointwise the sum of finitely many . In other words, for each , for some positive integer , for all . Thus the function is well defined and is continuous at each . Note that for each , , showing that it is unbounded.

The directions and are clear.

Let be a continuous function. We want to show that is a bounded function. Consider the open family where each . Note that is a locally finite family in since its members are inverse images of members of a locally finite family in the range space . By condition , has a finite subcover, leading to the conclusion that is a bounded function.

**Theorem 2**

Let be a space. The following conditions are equivalent:

- The space is pseudocompact.
- If is a family of non-empty open subsets of such that for each , then .
- If is a family of non-empty open subsets of such that has the finite intersection property, then .

**Proof**

Suppose that is pseudocompact. Suppose satisfies the hypothesis of condition . If there is some positive integer such that for all , then we are done. So assume that are distinct for infinitely many . According to condition in Theorem 1, must not be a locally finite family. Then there exists a point such that every open set containing must meet infinitely many . This implies that for infinitely many . Thus .

Suppose is a family of non-empty open sets with the finite intersection property as in the hypothesis of . Then let , , , and so on. By condition , we have , which implies .

Let be a continuous function such that is unbounded. For each positive integer , let . Clearly the open sets have the finite intersection property. Because is unbounded, it follows that .

Let be a space. Let . The interior of , denoted by , is the set of all points such that there exists an open set with . Points of are called the interior points of . A subset is said to be a closed domain if . It is clear that is a closed domain if and only if is the closure of an open set.

**Theorem 3**

The property of being a pseudocompact space is hereditary with respect to subsets that are closed domains.

**Proof**

Let be a pseudocompact space. We show that is pseudocompact for any nonempty open set . Let where is a non-empty open subset of . Let be a decreasing sequence of open subsets of . Note that each contains points of the open set . Let for each . Note that the open sets form a decreasing sequence of open sets in the pseudocompact space . By Theorem 2, we have (closure here is with respect to ). Note that points in are also points in (closure with respect to ). By Theorem 2, is pseudocompact.

* Theorem 4* (Statement above)

Let be a pseudocompact submetrizable space. Then is metrizable.

**Proof**

Let be a pseudocompact submetrizable space. Then there exists topology on such is metrizable and . We show that , leading to the conclusion that is also metrizable. If , we denote the closure of in by and the closure of in by .

To show that , we show any closed set with respect to the topology is also a closed set with respect to the topology . Let be a closed set in . Consider the family . We make the following claims.

* Claim 1*. .

* Claim 2*. Each is pseudocompact in .

* Claim 3*. Each is pseudocompact in .

* Claim 4*. Each is compact in .

We now discuss each of these four claims. For Claim 1, it is clear that . The reverse set inclusion follows from the fact that is a regular space. Claim 2 follows from Theorem 3. Note that sets in are closed domains in the pseudocompact space .

If sets in are pseudocompact in the larger topology , they would be pseudocompact in the weaker topology too. Thus Claim 3 is established. In a metrizable space, compactness and weaker notions such as countably compactness and pseudocompactness coincide. Because they are pseudocompact subsets, sets in are compact in the metrizable space . Thus Claim 4 is established.

It follows that is closed in since it is the intersection of compact sets in . Thus is identical to , implying that is metrizable.

*Reference*

- Engelking, R.,
*General Topology, Revised and Completed edition*, 1989, Heldermann Verlag, Berlin. - Willard, S.,
*General Topology*, 1970, Addison-Wesley Publishing Company.