It is a well known fact in topology that in any compact Hausdorff space, if is a point (it is the only point in the intersection of countably many open sets), then there is a countable local base at . See 3.1.F in  and 16.A.4 in . Thus for a compact Hausdorff space, if every point is a point, the space is first countable. In the previous post The cardinality of compact first countable spaces, I, we show that for every compact, Hausdorff space that is first countable, . Combining these two results, we have the following theorem:
If is a compact, Hausdorff and satisfies the condition that every point of is a point, then , i.e. the cardinality of is no greater than continuum (the cardinality of the real line).
In this post, we prove that in a compact Hausdorff space, there is a countable local base at every point that is a .
Let be a topological space. The space is said to be a Hausdorff space if for with , there exist disjoint open sets and such that and . Hausdorff spaces are also called T2 spaces. The space is said to be a regular space if it is T2 and for each and for each closed subset with , there exist disjoint open sets and such that and . Regular spaces are also called T3 spaces.
A local base at is a collection of open sets containing the point such that if where is open, then for some . The space satisfies the first axiom of countability if there is a countable local base at each point in the space. The space is said to be first countable if it satisfies the first axiom of countability.
If the space is regular and has a countable local base at , then we can have a countable local base at that satisfies the following:
Let . The point is said to be a point in if where each is open in . Clearly, if there is a countable local base at , then the point is a point. The converse is not true.
The space is said to be a compact space if every open cover of has a finite subcover. The space is said to be countably compact if every countable open cover of has a finite subcover. Equivalently, is countably compact if and only if every infinite has a limit point in .
Suppose is a regular countably compact space in which every point is a point. Then is first countable.
Proof. Let . We can find countably many open sets whose intersection is . Since the space is regular, the open sets can be made to satisfy the condition that .
We claim that for each open set containing , for some . Suppose not. Then there is an open set with such that each is not a subset of . For each , choose such that . The set must be infinite. Otherwise, there is some such that for all . Then since belongs to each (a contradiction).
Since is infinite, has a limit point . Note that for each the point is a limit point of
Thus for each . This mean and the open set would contain infinitely many . But is chosen to be not in , a contradiction. It follows that the open sets that make the point a point must be a local base.
Let be a compact T2 space such that every point is a point. Then is first countable.
Proof. This follows from theorem 1. Note that any compact T2 space is normal and is thus regular.
- Engelking, R. General Topology, Revised and Completed edition, 1989, Heldermann Verlag, Berlin.
- Willard, S., General Topology, 1970, Addison-Wesley Publishing Company.