The cardinality of compact first countable spaces, II

It is a well known fact in topology that in any compact Hausdorff space, if x \in X is a G_\delta point (it is the only point in the intersection of countably many open sets), then there is a countable local base at x. See 3.1.F in [1] and 16.A.4 in [2]. Thus for a compact Hausdorff space, if every point is a G_\delta 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 X that is first countable, \lvert X \lvert \le 2^\omega. Combining these two results, we have the following theorem:

Theorem
If X is a compact, Hausdorff and satisfies the condition that every point of X is a G_\delta point, then \lvert X \lvert \le 2^\omega, i.e. the cardinality of X 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 G_\delta.

Definitions
Let X be a topological space. The space X is said to be a Hausdorff space if for x,y \in X with x \ne y, there exist disjoint open sets U \subset X and V \subset X such that x \in U and y \in V. Hausdorff spaces are also called T2 spaces. The space X is said to be a regular space if it is T2 and for each x \in X and for each closed subset C \subset X with x \notin C, there exist disjoint open sets U \subset X and V \subset X such that x \in U and C \subset V. Regular spaces are also called T3 spaces.

A local base at x \in X is a collection \mathcal{B} of open sets containing the point x such that if x \in U \subset X where U is open, then x \in B \subset U for some B \in \mathcal{B}. The space X satisfies the first axiom of countability if there is a countable local base at each point in the space. The space X is said to be first countable if it satisfies the first axiom of countability.

If the space X is regular and has a countable local base at x \in X, then we can have a countable local base \left\{U_n:n=1,2,3,\cdots\right\} at x \in X that satisfies the following:

\overline{U_{n+1}} \subset U_n for each n=1,2,3,\cdots

Let x \in X. The point x is said to be a G_\delta point in X if \left\{x\right\}=\bigcap \limits_{n=1}^{\infty}U_n where each U_n is open in X. Clearly, if there is a countable local base at x \in X, then the point x is a G_\delta point. The converse is not true.

The space X is said to be a compact space if every open cover of X has a finite subcover. The space X is said to be countably compact if every countable open cover of X has a finite subcover. Equivalently, X is countably compact if and only if every infinite A \subset X has a limit point in X.

Theorem 1
Suppose X is a regular countably compact space in which every point is a G_\delta point. Then X is first countable.

Proof. Let x \in X. We can find countably many open sets U_n whose intersection is \left\{x\right\}. Since the space is regular, the open sets U_n can be made to satisfy the condition that \overline{U_{n+1}} \subset U_n.

We claim that for each open set U containing x, U_n \subset U for some n. Suppose not. Then there is an open set U with x \in U such that each U_n is not a subset of U. For each n, choose x_n \in U_n such that x_n \notin U. The set A=\left\{x_n:n=1,2,3,\cdots\right\} must be infinite. Otherwise, there is some j such that x_j=x_k for all k>j. Then x_j=x since x_j belongs to each U_n (a contradiction).

Since A is infinite, A has a limit point p. Note that for each j the point p is a limit point of

A_j=\left\{x_n:n=j,j+1,j+2,\cdots\right\} \subset U_j.

Thus p \in \overline{U_j} for each j. This mean p=x and the open set U would contain infinitely many x_n. But x_n is chosen to be not in U, a contradiction. It follows that the open sets U_n that make the point x a G_\delta point must be a local base. \blacksquare

Corollary 2
Let X be a compact T2 space such that every point is a G_\delta point. Then X is first countable.

Proof. This follows from theorem 1. Note that any compact T2 space is normal and is thus regular. \blacksquare

Reference

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