Is every ccc space weakly Lindelöf?

Question

A space is separable if it has a countable dense subset. A space has the countable chain condition (ccc) if every collection of pairwise-disjoint open sets is countable. Finally, a space is weakly Lindelöf if every open cover has a countable subcollection whose union is dense in the space.

Every separable space has the ccc.

A separable space is also weakly Lindelöf: choose an open set from the cover for each member of the countable dense subset to obtain the desired countable subcollection. This is currently Theorem T129 of the pi-Base.

Can T129 be improved to only assume ccc rather than separable?

Answer 1

Yes. A current pull request to the pi-Base observes this can be seen from chapter 1, section 3 of The Handbook of Set-Theoretic Topology. I'd like to provide a direct proof for reference.

Given a ccc space and an open cover $\mathcal U$, let's use transfinite induction on $\omega_1$. For each $\alpha<\omega_1$, we will choose an open set $U_\alpha$ which is a subset of some member of $\mathcal U$. If $\bigcup_{\beta<\alpha}U_\beta$ is dense in the space, let $U_\alpha=\emptyset$. Otherwise, pick $x_\alpha\not\in\overline{\bigcup_{\beta<\alpha}U_\beta}$, pick $V_\alpha\in\mathcal U$ with $x_\alpha\in V_\alpha$, and let $U_\alpha=V_\alpha\setminus \overline{\bigcup_{\beta<\alpha}U_\beta}\not=\emptyset$.

We claim that there exists some $\alpha<\omega_1$ such that $\bigcup_{\beta<\alpha}U_\beta$ is dense in the space, which shows the space is weakly Lindelöf. To see this, consider $\{U_\alpha:\alpha<\omega_1\}$. Since the space is ccc, and this collection is pairwise-disjoint, only countably-many $U_\alpha$ may be nonempty. Since a $U_\alpha$ is empty only when $\bigcup_{\beta<\alpha}U_\beta$ is dense in the space, we're done.

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For convenience, here is the generalized proof that every $\kappa$ chain condition ($\kappa$cc) space is weakly $\kappa$-Lindelöf.

Given a $\kappa$cc space and an open cover $\mathcal U$, let's use transfinite induction on $\kappa^+$. For each $\alpha<\kappa^+$, we will choose an open set $U_\alpha$ which is a subset of some member of $\mathcal U$. If $\bigcup_{\beta<\alpha}U_\beta$ is dense in the space, let $U_\alpha=\emptyset$. Otherwise, pick $x_\alpha\not\in\overline{\bigcup_{\beta<\alpha}U_\beta}$, pick $V_\alpha\in\mathcal U$ with $x_\alpha\in V_\alpha$, and let $U_\alpha=V_\alpha\setminus \overline{\bigcup_{\beta<\alpha}U_\beta}\not=\emptyset$.

We claim that there exists some $\alpha<\kappa^+$ such that $\bigcup_{\beta<\alpha}U_\beta$ is dense in the space, which shows the space is weakly $\kappa$-Lindelöf. To see this, consider $\{U_\alpha:\alpha<\kappa^+\}$. Since the space is $\kappa$cc, and this collection is pairwise-disjoint, only $\kappa$-many $U_\alpha$ may be nonempty. Since a $U_\alpha$ is empty only when $\bigcup_{\beta<\alpha}U_\beta$ is dense in the space, we're done.