Q:

A question about linearly lindelöf space

Suppose $X$ be a linearly lindelöf space, and $\mathcal{P}$ a family closed subsets of $X$, satisfying $\cap \mathcal{P}=\emptyset$. Let $\tau$ be the smallest cardinal number such that there is a subfamily $\gamma$ of $\mathcal{P}$ satisfying the conditions: $\cap \gamma=\emptyset$ and $|\gamma|=\tau$. Assume that $\tau$ is uncountable. How could I show the cofinality of $\tau$ is $\omega$? Thanks ahead:)

A:

Let $\gamma=\{F_\xi:\xi<\tau\}$. For $\eta<\tau$ let $H_\eta=\bigcap_{\xi\le\eta}F_\xi$, and let $U_\eta=X\setminus H_\eta$. Then $\mathscr{U}=\{U_\xi:\xi<\tau\}$ is a non-decreasing open cover of $X$, so it has a countable subcover $\{U_{\xi_n}:n\in\omega\}$, where $\xi_n<\xi_{n+1}$ for each $n\in\omega$. The sequence $\langle\xi_n:n\in\omega\rangle$ must be cofinal in $\tau$.

