How to prove that every first countable, ccc, Hausdorff space has cardinality at most $2^\omega$ by use the Erdős-Rado theorem?

Erdős-Rado theorem: Let $\kappa$ be an infinite cardinal. Let $E$ be a set with $|E|>2^\kappa$, and suppose $[E]^2=\bigcup_{\alpha<\kappa}P_\alpha$. Then there exists $\alpha<\kappa$ and a subset $A$ of $E$ with $|A|>\kappa$ such that $[A]^2\subset P_\alpha$.

Thanks ahead:)


Suppose that $|X|>2^\omega$. For each $x\in X$ let $\mathscr{B}(x)=\{B_n(x):n\in\omega\}$ be a countable local base at $x$; without loss of generality assume that $B_n(x)\supseteq B_{n+1}(x)$ for each $x\in X$ and $n\in\omega$. For $n\in\omega$ let

$$P_n=\left\{\{x,y\}\in[X]^2:n=\min\{k\in\omega:B_k(x)\cap B_k(y)=\varnothing\}\right\}\;;$$

$X$ is Hausdorff, so $\bigcup_{n\in\omega}P_n=[X]^2$. By the Erdős-Rado theorem there are an uncountable $A\subseteq X$ and an $n\in\omega$ such that $[A]^2\subseteq P_n$. But then $\{B_n(x):x\in A\}$ is an uncountable pairwise disjoint family of non-empty open sets, which is impossible.

By the way, essentially the same argument shows that if $X$ is Hausdorff, then

$$|X|\le 2^{c(X)\chi(X)}\;,$$

where $c(X)$ is the cellularity and $\chi(X)$ the character of $X$. (These cardinal functions are defined here.)


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