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 space.

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).