more examples of non-weakly Lindelöf spaces

Question

A space $X$ is called weakly-Lindelöf if every open cover $\mathcal{U}$ has a countable subcover $\mathcal{U'} \subseteq \mathcal{U}$ such that $\cup \mathcal{U}'$ is dense in $X$.

This class seems to be of interest in functional analysis (spaces of continuous functions). Every Lindelöf space or space with a dense Lindelöf subspace (like separable spaces) is weakly-Lindelöf.

In this paper by Hajnal and Juhasz they construct a ZFC example of two Lindelöf spaces whose product is not weakly Lindelöf (a variation on the Sorgenfrey line construction). Besides this example and the obvious example of an uncountable discrete space, I've not been able to find non-examples, i.e. spaces that are not weakly-Lindelöf. Does anyone have any pointers to where to find more of those? To understand a class of spaces I prefer to have both examples and non-examples.

This paper by Frolik is supposed to have defined this notion originally, but I cannot read Russian, so I cannot check. Maybe the "seminal paper" contains more examples, I thought.

Answer 1

It is not hard to show that a weakly-Lindelof metric space is separable (basically the same proof used for Lindelof spaces works). Hence $l^\infty$ or your favorite non-separable metric space is an example of a non-weakly-Lindelof.

Another example is $\omega_1$ with the order topology (which is also first-countable but non-metrizable), or any LOTS with uncountable cofinality (just consider the open cover of initial segments).

Answer 2

More generally, any non-Lindelof paracompact space (which includes any non-separable metric space) is an example of a non-weakly Lindelof space.

This follows immediately from Taras Banakh's observation in another post that every locally finite family of open sets in a weakly Lindelof space is at most countable.