Let $X$ be a $T_1$ space and let $F[X]$ be $\{x\subset X:\text{is finite}\}$ with Pixley-Roy topology.
If $X$ is not discrete, how to prove $F[X]$ is not a Baire space?
Thanks ahead:)
Definition of Pixley-Roy topology: Basic neighborhoods of $F\in F[X]$ are the sets $[F,V]=\{H\in F[X]; F\subseteq H\subseteq V\}$ for open sets $V\supseteq F$, see e.g. here.
I don't know in the theorem 2.2 why each $Z \cap F_n[X]$ is closed, nowhere dense subspace of $Z$?