a question on Pixley-Roy topology

Question

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:)

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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.

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I don't know in the theorem 2.2 why each $Z \cap F_n[X]$ is closed, nowhere dense subspace of $Z$?

Answer 1

(explaining the theorem 2.2 in the paper linked by Martin Sleziak)

Let $X$ not be discrete, and $p$ be a limit point.

Put $F_n[X]=\lbrace A\subseteq X\vert \lvert A\rvert\leq n\rbrace$, $Z=[\lbrace p\rbrace,X]$

If $F[Z]$ were Baire, so would $Z$. But $Z=\bigcup_n (Z\cap F_n[X])$, and each of $Z\cap F_n[Z]$ is nowhere dense and closed:

• it is closed because its complement $\bigcup_{\lvert A\rvert >n} (Z\cap[A,X])$ is open.
• it is nowhere dense, because any nonempty basic open subset of $Z$ is of the form $[A,U]$ for $p\in A\subseteq U$, so in particular $U$ is infinite (as a neighbourhood of $p$), so it has an $n+1$-element subset $B$. Then $[A\cup B,U]$ is a subset of $[A,U]$ disjoint from $F_n[X]\cap Z$