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A collection $\{X_\alpha : \alpha\in\Lambda\}$ is said to be a $\omega$-cover of a space $X$ if for each finite $F\subseteq X$ there exists a $\beta\in\Lambda$ such that $F\subseteq X_\beta$. ($\{X_\alpha : \alpha\in\Lambda\}$ is called an open $\omega$-cover of $X$ if each $X_\alpha$ is open in $X$.)

Let $\{X_\alpha : \alpha\in\Lambda\}$ (not necessarily countable) be a $\omega$-cover of a space $X$. If for each $\alpha$ $Y_\alpha$ is a dense subset of $X_\alpha$, then for what condition on $\{X_\alpha : \alpha\in\Lambda\}$, $\{Y_\alpha : \alpha\in\Lambda\}$ is a $\omega$-cover of $\cup_{\alpha\in\Lambda}Y_\alpha$.

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    $\begingroup$ On line 4, is "a $\omega$-cover" supposed to be "an open $\omega$-cover"? (If not, I don't know why you bothered to define open $\omega$-cover.) And what sort of conditions do you have in mind? Conditions on $X$? Conditions on $\{X_\alpha\}$? Conditions on $\{Y_\alpha\}$? What sort of conditions will rule out trivial counterexamples like $X_\alpha=X$ and $Y_\alpha$ pairwise disjoint dense sets? $\endgroup$
    – bof
    Apr 24 at 23:27

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