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P000243 |
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Has countable $\pi$-weight |
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| zb |
name |
0559.54003 |
Cardinal functions I (R. Hodel), Ch. 1 of Handbook of set-theoretic topology |
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$X$ has a countable $\pi$-base.
A $\pi$-base for $X$ is a collection $\mathcal{V}$ of nonempty open sets in $X$ such that every nonempty open $U\subseteq X$ contains some $V\in\mathcal V$.
Such a space has countable $\pi$-weight, where the $\pi$-weight is defined as
$\quad\quad\pi w(X):=\min{|\mathcal V|:\mathcal V \text{ is a }\pi\text{-base for }X}+\omega.$
Defined on page 14 of {{zb:0559.54003}}.
- This property is hereditary with respect to dense sets.
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$X$ satisfies this property iff its Kolmogorov quotient $\operatorname{Kol}(X)$ does.
- This property is preserved by countable products.