# Dense refinement of $\omega$-cover

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 $$\cup_{\alpha\in\Lambda}Y_\alpha$$.

• 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?
– bof
Apr 24 at 23:27