$X$ is a complete metric space, $Y$ is compact. $X \times Y$ is Baire?

Question

Requesting a hint or solution.

X is a complete metric space and Y is a compact hausdorff space. Trying to show that $X \times Y$ is a Baire space.

Answer 1

$\newcommand{\cl}{\operatorname{cl}}$You need to show that if $G_n$ is a dense open subset of $X\times Y$ for each $n\in\Bbb N$, then $G=\bigcap_{n\in\Bbb N}G_n$ is dense in $X\times Y$. You can do this by imitating the proofs of the Baire category theorem for complete metric spaces and compact Hausdorff spaces simultaneously.

Without loss of generality we may assume that $G_0\supseteq G_1\supseteq G_2\supseteq\ldots\;$. (Why?) Let $U$ be a non-empty open set in $X\times Y$. $G_0$ is dense in $X\times Y$, so we can pick a point $p_0\in U\cap G_0$, say $p_0=\langle x_0,y_0\rangle$. Choose open nbhds $V_0$ of $x_0$ and $W_0$ of $y_0$ such that $\cl_XV_0\times\cl_YW_0\subseteq U\cap G_0$ and $\operatorname{diam}(\cl_XV_0)\le 1$, and let $U_1=V_0\times W_0$. Given $U_n$, choose $p_n=\langle x_n,y_n\rangle\in U_n\cap G_n$ and proceed similarly, but make $\operatorname{diam}(\cl_XV_n)\le 2^{-n}$. Can you finish it from here?