# 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.