Are W-spaces with countable pseudocharacter first countable?

Question

Before stating my question (title apart) I am going to introduce the following definition:

A topological space $X$ is called W-space if P1 has a winning strategy at each point $x \in X$ for the following game defined by Gruenhage in DOI 10.1016/0016-660X(76)90024-6.

During each round $n < \omega$, P1 chooses some neighborhood $U_n$ of $x$, then P2 chooses some point $p_n \in U_n$. P1 wins provided the sequence $p_n$ converges to $x$; P2 wins otherwise.

I was wondering if W-spaces with countable pseudocharacter (each point is a countable intersection of open sets) are first countable, or if anyone knows an example of a W-space with countable pseudocharacter that isn't first countable.

If it helps, note that in spaces with countable pseudocharacter, being well-based is equivalent to being first countable (see here).

Note: there are no examples verifying W-space+Has points $G_\delta$+~First countable in $\pi$-Base yet. However, there are candidates, which all are of the following form: ?W-space+Has points $G_\delta$+~First countable (it isn't deduced yet whether they verify the W-space property or not).

Answer 1

Moving from the comments to write up details on the status of the W-space property for a few of those spaces found in π-Base.

S131 Sequential fan* is not W: during round $n$, P2 may choose any point within P1's open set that is on the $n$th blade of the fan. The result is a sequence of points $\langle n,f(n)\rangle_{n<\omega}$ for some $f:\omega\to\omega$, which will be missed by the open neighborhood $U_{f+1}$ of $\infty$.

Then S139 Bouquet of circles, S97 Steen-Seebach maximal compact topology, and S186 Converging sequence of non-Hausdorff spaces all fail to be W-spaces, because they all contain a copy of S131.

Finally, S29 One Point Compactification of the Rationals is not W: for the compactifying point, note that every neighborhood of this point is dense in $\mathbb Q$. For any strategy of P1, during round $n$, P2 may legally choose some point $x_n\in [0,2^{-n})$. As a result, $\{0\}\cup\{x_n:n<\omega\}$ is a compact subset of $\mathbb Q$ and its complement is a neighborhod of the compactifying point, showing that P2's sequence fails to converge.

Sadly, your question remains open, but at least this prunes the search space for counterexamples a bit.

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In https://math.stackexchange.com/a/1644092/86887 it's shown that a countable box product of reals is not first-countable, but does have points $G_\delta$ (countable pseudocharacter). π-Base is missing points $G_\delta$ as of this post.

I claim this space is not $W$, or even $w$. (A $w$ space is where the second player lacks a winning strategy, which is weaker than $W$ due to indeterminacy of infinite-length games.) Let $U_f=\prod_{n<\omega}(-f(n),f(n))$ and note $\{U_f|f:\omega\to\mathbb R^+\}$ is a local base for $\vec 0$ in the box topology on $\prod_{n<\omega}\mathbb R$. So we may define the tactical strategy (aka stationary strategy, which considers only the most recent move of the opponent) $\sigma(U_f)=f/2$ for P2 in the $W$-game at $\vec 0$, and we will show this strategy is winning.

Given any attack $U_{f_0},U_{f_1},\dots$ of P1 (WLOG using basic open sets) against $\sigma$, define $g:\omega\to\mathbb R^+$ by $g(n)=f_n(n)/4$ and consider $U_g$. For each $n<\omega$, we have $\sigma(U_{f_n})(n)=f_n(n)/2>f_n(n)/4=g(n)$, and thus $\sigma(U_{f_n})\not\in U_g$. It follows that $U_g$ is a neighborhood of $\vec 0$ missing every move of P2, showing that $\sigma$ is a winning tactical strategy for P2 in the $W$-game.

EDIT: Or just simply note that this space contains a copy of S131: let $\vec 0$ be the particular point, then take a converging sequence along each coordinate (setting all other coordinates zero).

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* $X=(\omega\times\omega)\cup\{\infty\}$, with all the points of $\omega\times\omega$ isolated and neighborhoods of $\infty$ being the sets omitting finitely many points from each of the columns of $\omega\times\omega$. Basic open neighborhoods of $\infty$ are given by the sets $U_f=\{\infty\}\cup\{\langle m,n\rangle\in\omega\times\omega:n\ge f(m)\}$ for some arbitrary function $f:\omega\to\omega$.

Answer 2

The weak topology on a separable Hilbert space (S21) and Van Douwen's anti-Hausdorff Fréchet space (S161) are out of the list too.

The weak topology on a separable Hilbert space is separable, regular (it has a group topology), and not first countable (it is $T_2$ (proof) and has a group topology, but it isn't metrizable). By the theorem $3.6$ of DOI 10.1016/0016-660X(76)90024-6, we deduce that this space isn't a W-space.

Van Douwen's anti-Hausdorff Fréchet space is countable, and not first countable (it is hyperconnected, not finite, and every convergent sequence has exactly one limit). By the theorem $3.3$ of DOI 10.1016/0016-660X(76)90024-6, we deduce that this space isn't a W-space.