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If $\kappa$ is infinite cardinal and $X$ is a topological space with $\operatorname{w}(X)\leq\kappa$, then, for every base $\mathcal{B}$ of $X$, there exists an $\mathcal{B}_0\subseteq \mathcal{B}$ such that $\mathcal{B}_0$ is a base for $X$ and $|\mathcal{B}_0|\leq\kappa$. The proof of this result is quite similar for network. I would like to know if the result holds for $\pi$-base

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Paul’s given you the answer; here’s a proof. Let $\kappa=\pi w(X)$, and let $\mathscr{P}$ be a $\pi$-base for $X$ such that $|\mathscr{P}|\le\kappa$. Let $\mathscr{B}$ be any $\pi$-base for $X$. For each $P\in\mathscr{P}$ there is a $B_P\in\mathscr{B}$ such that $B_P\subseteq P$. Let $\mathscr{B}_0=\{B_P:P\in\mathscr{B}\}$; clearly $|\mathscr{B}_0|\le\kappa$. Moreover, if $U$ is any non-empty open set in $X$, there is some $P\in\mathscr{P}$ such that $P\subseteq U$ and hence $B_P\subseteq P\subseteq U$, so $\mathscr{B}_0$ is a $\pi$-base for $X$.

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The answer is Yes. A $\pi$-base for $X$ is a collection $\mathscr V$ of non-empty open sets in $X$ such that if $R$ is any non-empty open set in $X$, then $V \subseteq R$ for some $V \in \mathscr V$.

Example: $\{\{n\}: n\in \omega\}$ is a $\pi$-base for $\beta \omega$.

The $\pi$-weight of $X$ is defined as follows: $$\pi w(X)=\min\{|\mathscr V|: \mathscr V \text{ a } \pi-base \text{ for } X\}+\omega .$$

Note that $d(X)\le\pi w(X)\le w(X)$. The cardinal function $\pi$-weight is not monotone; e.g., $\pi w(\beta \omega)=\omega$ but $\pi w(\beta \omega \setminus \omega)=2^\omega$.

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