# Is there a locally countable and weakly Lindelöf space which is not ccc

Is there a locally countable and weakly Lindelöf space which is not ccc?

A space $$X$$ is locally countable if for each point $$x\in X$$ there is an open neighbourhood $$O_x$$ of $$x$$ such that $$|O_x| < \omega_1$$;

Here ccc denotes the countable chain condition;

A space $$X$$ is called weakly Lindeöf if for any open cover $$\mathcal U$$ of $$X$$ there is a counable subset $$\mathcal V \subset \mathcal U$$ such that $$\bigcup \mathcal V$$ is dense in $$X$$.

Thank you.

No, because "locally countable + weakly Lindelöf" $$\Rightarrow$$ separable $$\Rightarrow$$ ccc.
For the first implication, use local countability to choose for each $$x \in X$$ some countable open neighborhood $$O_x \ni x$$. Then $$\mathcal U = \{O_x \,:\, x \in X\}$$ is an open cover of $$X$$, and we may use the weakly Lindelöf property to find a countable $$\mathcal V \subseteq \mathcal U$$ such that $$\bigcup \mathcal V$$ is dense in $$X$$. But $$\bigcup \mathcal V$$ is countable, so this shows $$X$$ is separable.
For the second implication, let $$D$$ be any countable dense subset of $$X$$. If $$\mathcal U$$ is a collection of pairwise disjoint open subsets of $$X$$, then each $$U \in \mathcal U$$ contains some $$d_U \in D$$ (because $$U$$ is open) and the function $$U \mapsto d_U$$ is an injection (because of "pairwise disjoint"). Thus $$\mathcal U$$ is countable.