In this blog I have already presented two metrization theorems for compact spaces: (1) any compact space with a countable network is metrizable (see the post), (2) any compact space with a diagonal is metrizable (see the post). I now present another classic theorem: any countably compact space with a point-countable base is metrizable. This theorem is a classic result of Miscenko ([1]). All spaces are at least Hausdorff and regular. We have the following three metrization theorems for compact spaces. In subsequent posts, I will discuss generalizations of these theorems and discuss related concepts.

**Thoerem 1**. Any compact space with a countable network is metrizable.

The proof is in this post.

**Thoerem 2**. Any compact space with a is metrizable.

The proof is in this post.

**Thoerem 3**. Any countably compact space with a point-countable base is metrizable.

A base for a space is a point-countabe base if every point in belongs to at most countably elements of .

**Proof of Theorem 3**. Let be a point-countable base for the countably compact space . We show that is separable. Once we have a countable dense subset, the base has to be a countable base. So we inductively define a sequence of countable sets such that is dense in .

Let be a one-point set to start with. For , let . Let . For each finite such that , choose a point . Let be the union of and the set of all points . Let .

We claim that . Suppose we have . Let . We know that is countable since every element of contains points of the countable set . We also know that is an open cover of . By the countably compactness of , we can find a finite such that . The finite set must have appeared during the induction process of selecting points for for some (i.e. ). So a point has been chosen such that (thus we have ). On the other hand, since , we observe that , producing a contradiction. Thus the countable set is dense in , making the point-countable base a countable base.

**Reference**

- Miscenko, A.,
*Spaces with a point-countable base*, Dokl. Acad. Nauk SSSR, 144 (1962), 985-988. (English translation: Soviet Math. Dokl. 3 (1962), 1199-1202).

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