Example of $\pi$-metrizable space

Question

A tychonoff space $X$ is $\pi$-metrizable if and only if it has a $\sigma$-locally finite $\pi$-base.

Please help me to find some example of $\pi$-metrizable space.

Is it true that every $\pi$-metrizable space is a moscow space ?

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For a space $X$,

A collection of nonempty open sets $\mathcal{A}$ , is called a $\pi$-base if for every nonempty open set $O$, there exists $U \in\mathcal{A}$ such that $U\subset O$.

Answer 1

Theorem $3.4$ of Derrick Stover, ‘On $\pi$-metrizable spaces, their continuous images and products’, says that if $X$ is any Tikhonov space, and $D$ is the discrete space of cardinality $\pi w(X)$, then $X\times D^\omega$ is $\pi$-metrizable; that provides quite a few examples. In particular, let $\kappa$ be any infinite cardinal, let $D$ be the discrete space of cardinality $\kappa$, and let $X=D^\kappa$. Then $d(X)=\kappa$ by the Hewitt-Marczewski-Pondiczery theorem, so $\pi w(X)=\kappa$, and $X\cong X\times D^\omega$ is $\pi$-metrizable.

However, it follows from Corollary $2.12$ of A. V. Arhangel’skiǐ, Moscow spaces and topological groups, Top. Proc. 25 (2000), 383-416, that if $X\times D^\omega$ is Moscow, then so is $X$. Thus, $X\times D^\omega$ is a $\pi$-metrizable space that is not Moscow if $X$ is not Moscow.