$X\times Y$ is second countable Baire spaces

Question

If $X,Y$ are second countable Baire spaces, so is $X\times Y$. ?

If true, a suggestion to try thanks

Answer 1

Yes. It suffices $X$ to be second countable for $X\times Y$ be Baire, by Theorem 5.1.viii from [HC].

[HC] R. C. Haworth, R. C. McCoy, Baire spaces, Warszawa, Panstwowe Wydawnictwo Naukowe, 1977.

Answer 2

HINT: Let $\mathscr{B}$ be a countable base for $Y$. Let $\pi_X:X\times Y\to X$ be the usual projection map. Suppose that $G_n$ is a dense open subset of $X\times Y$ for each $n\in\Bbb N$.

• Show that $\pi_X[G_n\cap(X\times B)]$ is a dense open subset of $X$ for each $n\in\Bbb N$ and $B\in\mathscr{B}$.

Let $D_n=\bigcap_{B\in\mathscr{B}}\pi_X[G_n\cap(X\times B)]$.

• Show that each $D_n$ is dense in $X$.
• Show that if $x\in D_n$, then $\{y\in Y:\langle x,y\rangle\in G_n\}$ is a dense open subset of $Y$.

Let $D=\bigcap_{n\in\Bbb N}D_n$.

• Show that $D$ is dense in $X$.
• Show that if $x\in D$, then $\left\{y\in Y:\langle x,y\rangle\in\bigcap_{n\in\Bbb N}G_n\right\}$ is dense in $Y$.
• Show that $\bigcap_{n\in\Bbb N}G_n$ is dense in $X\times Y$.

(Note that I did not need to assume that $X$ was second countable.)