How to prove that $|X|\le 2^{s(X)\psi(X)}$ by using the Erdős-Rado theorem when $s(X)=\psi(X)=\omega$?

$s(X)=\sup \{ |D|: D \subset X, D \text{ is discrete} \} + \omega $

$\psi(X)= \sup\{\psi(p,X): p \in X\} + \omega$, $\psi(p,X)=\min \{ |\mathcal V|: \mathcal V \text{ is a pseudo-base for } p \}$

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 $X$ is $T_1$, and $\psi(X)\le\kappa$. Suppose further that $|X|>2^\kappa$; I’ll show that $X$ has a discrete subset of cardinality $\kappa^+$.

Let $\preceq$ be any linear order on $X$, and for each $x\in X$ let $\mathscr{V}_x=\{V_\xi(x):\xi<\kappa\}$ be a family of open nbhds of $x$ such that $\bigcap\mathscr{V}_x=\{x\}$. Now partition $[X]^2$ as follows: for each $\langle\xi,\eta\rangle\in\kappa\times\kappa$ let

$$\mathscr{I}(\xi,\eta)=\left\{(x,y)_\preceq:x\prec y\text{ and }y\notin V_\xi(x)\text{ and }x\notin V_\eta(y)\right\}\;.$$

Of course $|\kappa\times\kappa|=\kappa$, so the Erdős-Rado theorem applies to gives us a $D\subseteq X$ and $\langle\xi,\eta\rangle\in\kappa\times\kappa$ such that $|D|=\kappa^+$, and $[D]^2\subseteq\mathscr{I}(\xi,\eta)$. Let $x\in D$; then clearly

$$V_\xi(x)\cap V_\eta(x)\cap D=\{x\}\;,$$

and $D$ is indeed discrete. Thus, $s(X)\ge\kappa^+$. Taking the contrapositive, we have

$$|X|\le 2^{s(X)\psi(X)}$$

for all $T_1$-spaces $X$.


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