It is a well known result that any countably compact and metacompact space is compact (see Theorem 5.3.2 in [1]). A discussion of this result is also found in this blog (countably compact + metacompact). Since countably compactness implies pseudocompactness, a natural question arises: can this result be generalized to “any pseudocompact and metacompact space is compact?” The answer is yes and was established in [2] and [3]. In this post, we put together a proof of this result by using building blocks already worked out in this blog.

All spaces considered here are Tychonoff (completely regular). Refer to [1] and [4] for any terms and notions not defined here (or refer to elsewhere in this blog).

A space is said to be almost compact if for every open cover of , there is a finite such that is dense in . It can be shown that for any regular space, almost compactness implies compactness. We have the following lemma.

**Lemma 1**

Let be a regular space. Then is compact if and only if is almost compact.

Theorem 2, Theorem 3 and Theorem 4 are building blocks proved in previous posts. Theorem 5 below is the main theorem. Corollary 6, the intended result, is obtained from applying Theorem 5 and Lemma 1.

* Theorem 2* (see Theorem 4B in this post)

Every regular pseudocompact is a Baire space.

* Theorem 3* (see Main Theorem in this post)

Let be a space. The following conditions are equivalent.

- is a Baire space.
- For any point-finite open cover of , the set is a dense set in .

* Theorem 4* (see Theorem 1 in this post)

Let be a space. The following conditions are equivalent.

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

**Theorem 5**

Let be a pseudocompact and metacompact space. Then is almost compact.

**Proof**

Let be an open cover of . By metacompactness, there is a which is a point-finite open refinement of . It suffices to find a finite such that covers a dense set.

By Theorem 2, is a Baire space. By Theorem 3, the set is dense in where . Let be the collection of all such that . Note that is open and dense in . Furthermore, it is straightforward to show that is locally finite at each point . By Theorem 4, is finite.

**Corollary 6**

Let be a pseudocompact and metacompact space. Then is compact.

*Reference*

- Engelking, R.,
*General Topology, Revised and Completed edition*, Heldermann Verlag, Berlin, 1989. - Scott, B., M.,
*Pseudocompact Metacompact Spaces are Compact*, Topology Proc., 4, 577-587, 1979. - Watson, W. S.,
*Pseudocompact Metacompact Spaces are Compact*, Proc. Amer. Math. Soc., 81, 151-152, 1981. - Willard, S.,
*General Topology*, Addison-Wesley Publishing Company, 1970.