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:

Countably Compact Spaces with G-delta Diagonals
Metrization Theorems for Compact Spaces

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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