# A note on metrization theorems for compact spaces

In a previous post (Metrization Theorems for Compact Spaces), three classic metrization theorems for compact spaces are discussed. The three theorems are: any Hausdorff compact space $X$ is metrizable if any of the following holds:

1. $X$ has a countable network,
2. $X$ has a $G_\delta$ diagonal,
3. $X$ has a point countable base.

The metrization results for conditions 2 and 3 hold for countably compact spaces as well. See the following posts:

In this post, we discuss another metrization theorem for compact spaces. We show that a compact Hausdorff space $X$ is metrizable if and only if the function space $C_p(X)$ is separable.

Let’s discuss the function space. Let $Y$ by any Tychonoff space and let $\mathbb{R}$ be the set of all real numbers. Let $C(Y,\mathbb{R})$ be the set of all real-valued continuous functions defined on $Y$. For any $A \subset Y$ and for any $V \subset \mathbb{R}$, define $[A,V]=\lbrace{f \in C(Y,\mathbb{R}): f(A) \subset V}\rbrace$. If we restrict $A$ to $\lbrace{x}\rbrace$ and restrict $V$ to open sets, then the set of all $[A,V]$ is a subbase for a topology on $C(Y,\mathbb{R})$. This topology is called the pointwise convergence topology. The function space $C(Y,\mathbb{R})$ with this topology is denoted by $C_p(Y)$.

It is a theorem that $C_p(Y)$ is separable if and only if $Y$ has a weaker topology that forms a separable metric space. The result on compact spaces is a corollary of this theorem.

Theorem. Let $Y$ be a Tychonoff space with $\tau$ being the topology. The following conditions are equivalent:

1. $C_p(Y)$ is separable.
2. There is a topology $\tau_1 \subset \tau$ such that $(Y, \tau_1)$ is a separable metric space.

Proof
$1 \Rightarrow 2$. Let $D \subset C_p(Y)$ be a countable dense subspace. Let $\mathcal{V}$ be the class of all bounded open intervals of $\mathbb{R}$ with rational endpoints. Consider $\mathcal{B}=\lbrace{f^{-1}(V): f \in D, V \in \mathcal{V}}\rbrace$. Note that $\mathcal{B}$ is a subbase for a topology $\tau_1$ on $Y$. Since $\mathcal{B}$ is countable, the topology $\tau_1$ has a countable base and is thus separable and metrizable.

$2 \Rightarrow 1$. Let $\tau_1 \subset \tau$ be a topology. Suppose that $\tau_1$ is generated by a countable base $\mathcal{U}$. As in $1 \Rightarrow 2$, let $\mathcal{V}$ be the class of all bounded open intervals of $\mathbb{R}$ with rational endpoints. Let $\mathcal{N}$ be the class of all finite intersections of the sets in the following collection of sets.

$\displaystyle \lbrace{[A,V]: A \in \mathcal{U},V \in \mathcal{V}}\rbrace$

Note that $\mathcal{N}$ is countable. For each $W \in \mathcal{N}$, choose $f_W \in W$. We claim that $\lbrace{f_W: W \in \mathcal{N}}\rbrace$ is a countable dense set of $C_p(Y)$. To see this, let $T=\bigcap_{i \le n} [A_i,V_i]$ be a basic open set in $C_p(Y)$ where $A_i=\lbrace{x_i}\rbrace$. Fix $f \in T$. For each $i$, choose $O_i \in \mathcal{U}$ such that $x_i \in O_i$ and $f(O_i) \subset V_i$. Then $W=\bigcap_{i \le n} [O_i,V_i] \in \mathcal{N}$. Now, we have $f_W \in T$.

Corollary. Let $X$ be a compact Hausdorff space. Then the following conditions are equivalent:

1. $X$ is metrizable.
2. $C_p(X)$ is separable.

Proof.
$1 \Rightarrow 2$. This follows from $2 \Rightarrow 1$ in the above theorem.

$2 \Rightarrow 1$. This follows from $1 \Rightarrow 2$ in the above theorem. Note that any compact Hausdorff space cannot have a strictly weaker (or coarser) Hausdorff topology. Thus if a compact Hausdorff space has a weaker metrizable topology, it must be metrizable.

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