A question on $P$-space

Question

A space $X$ called $P$-space if every $G_\delta$ subset of $X$ is open in $X$. Every discrete space, I believe, is $P$-space. My question is this: Could someone offer some other classical examples of $P$-space? Thanks ahead:)

Answer 1

Here is one very nice $P$-space that I think is not mentioned in the Misra paper noted by Arthur Fischer. Let $D=\{0,1\}$ be the discrete $2$-point space, and let $Y=D^{\omega_1}$ as a set. For $x\in Y$ let $\operatorname{supp}(x)=\{\xi<\omega_1:x_\xi=1\}$, the support of $x$. Let $X=\{x\in Y:|\operatorname{supp}(x)|<\omega\}$, the set of points with finite support. (Such a space is sometimes called a $\sigma$-product; is we keep the points with countable support, we have a $\Sigma$-product.)

For each countable partial function $\sigma$ from $\kappa$ to $D$ let $$B(\sigma)=\{x\in X:x\upharpoonright\operatorname{dom}\sigma=\sigma\}=\{x\in Y:\sigma\subseteq x\}\;,$$ and let $\mathscr{B}$ be the collection of such sets $B(\sigma)$; then $\mathscr{B}$ is a clopen base for a topology $\tau$ on $X$.

Proposition. $\langle X,\tau\rangle$ is a hereditarily paracompact, Lindelöf $P$-space.

Proof. For each $x\in X$ the family $\{B(y\upharpoonright\eta):\eta<\omega_1\}$ is a local base at $x$ well-ordered in type $\omega_1$ by $\supseteq$, so $X$ is a $P$-space.

Now let $\mathscr{B}_0=\{B(\sigma)\in\mathscr{B}:\operatorname{dom}\sigma=\eta\text{ for some }\eta<\omega_1\}$; this is still a base for $\tau$, and $\langle\mathscr{B}_0\supseteq\rangle$ is a tree. Let $V$ be any non-empty open set in $X$, and let $\mathscr{U}$ be an open cover of $V$; assume without loss of generality that $\bigcup\mathscr{U}=V$. Let $\mathscr{R}=\{B\in\mathscr{B}_0:B\subseteq U\text{ for some }U\in\mathscr{U}\}$, and let $\mathscr{R}_0$ be the set of $\supseteq$-minimal members of $\mathscr{R}$; then $\mathscr{R}_0$ is a clopen partition of $V$ that refines $\mathscr{U}$. In particular, $\mathscr{R}_0$ is locally finite, so $V$ is paracompact. It’s well-known that a space is hereditarily paracompact iff each of its open subsets is paracompact, so $X$ is hereditarily paracompact.

Finally, suppose that $X$ is not Lindelöf. Then it has an open cover $\mathscr{U}=\{U_\xi:\xi<\omega_1\}$ such that $U_\eta\setminus\bigcup_{\xi<\eta}U_\xi\ne\varnothing$ for each $\eta<\omega_1$. For each $\eta<\omega_1$ fix $x_\eta\in U_\eta\setminus\bigcup_{\xi<\eta}U_\xi$. For $\xi<\omega_1$ let $F_\xi=\operatorname{supp}(x_\xi)$. By the $\Delta$-system lemma there are an uncountable $\Lambda\subseteq\omega_1$ and a finite $F\subseteq\omega_1$ such that $\{F_\xi:\xi\in\Lambda\}$ is a $\Delta$-system with common part $F$. (That is, if $\xi,\eta\in\Lambda$ and $\xi\ne\eta$, then $F_\xi\cap F_\eta=F$.) Let $z$ be the unique element of $X$ whose support is $F$. Let $B(\sigma)$ be a basic open nbhd of $z$, and let $\alpha=\sup\operatorname{dom}\sigma$. Then for each $\xi\in\Lambda$ with $\min(F_\xi\setminus F)>\alpha$ we have $x_\xi\in B(\sigma)$, and there are uncountably many such $\xi$. On the other hand, $z\in U_\eta$ for some $\eta<\omega_1$, so $U_\eta$ is an open nbhd of $z$ disjoint from $\{x_\xi:\xi>\eta\}$. This is a contradiction, so $X$ is Lindelöf. $\dashv$

Answer 2

Take an uncountable set with the cocountable topology. Th intersection of countably many open sets is still cocountable, and so open.

Answer 3

The article

Misra, Arvind K., A topological view of $P$-spaces, General Topology and Appl. 2, pp.349–362

provides at least a few extra examples of P-spaces, as well as some topological facts.