I’m trying to prove the following claim:
“If $(X, \tau)$ is a Baire space, then $X \times [0,1]_{std}$ is a Baire space”
The only way I know how to prove this is by using a Theorem from John C Oxtoby, which states the following:
“If X and Y are Baire spaces and at least one of X and Y has a locally countable pseudo-base, then $X \times Y$ is a Baire space.” (Found in “Cartesian Products of Baire Spaces” by J.C. Oxtoby)
I would like to know if anyone know a proof for why $X \times [0,1]_{std}$ is a Baire space that doesn’t rely on the theorem by Oxtoby.
Theorem. (Kuratowski, Ulam) If $X, Y$ are Baire and $Y$ is second-countable then $X\times Y$ is Baire.
The above theorem is exercise $3.9.J$ c) in Engelking's General Topology.
They give the following hint:
It suffices to prove that every sequence $(G_n)_{n\in\mathbb{N}}$ of open dense subsets of $X\times Y$ has non-empty intersection. Take a countable basis $(U_n)_{n\in\mathbb{N}}$ of $Y$ consisting of non-empty sets, and show that the projection of $G_i\cap (X\times U_j)$ onto $X$ is an open and dense set for each $i, j$. Since $X$ is Baire we can take some $x_0\in \pi\left[\bigcap_{i, j} G_i\cap (X\times U_j)\right]$ where $\pi:X\times Y\to X$ is the projection onto $X$. The sets $H_i = \{y\in Y : (x_0, y)\in G_i\}$ are open and dense in $Y$ so again we can take some $y_0\in\bigcap_i H_i$ since $Y$ is Baire. Then $(x_0, y_0)\in\bigcap_i G_i$.