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nowhere dense subset; Booth's Lemma; $\pi $-weight
The following statement is proved to be independent from $[\operatorname{LB}+\neg \operatorname{CH}]$: \linebreak $(*)$ Let $X$ be a Tychonoff space with $c(X)\leq \aleph _0$ and $\pi w(X)<\frak C$. Then a union of less than $\frak C$ of nowhere dense subsets of $X$ is a union of not greater than $\pi w(X)$ of nowhere dense subsets.
[1] Rudin M.E.: Martin's Axiom. in Handbook of set-theoretic topology K. Kunen and J.E. Vaughan Elsevier Science Publishers B.V. (1984), 491-501.
[2] Bell M.G.: On the combinatorial Principle $P({\frak C})$. Fund. Math. 114 (1981), 149-157. MR 0643555 | Zbl 0581.03038
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