A weakly Lindelöf space which does not have the property ${}^{\ast }{\mathcal{U}}_{fin}(\mathcal{O},\mathcal{O})$

We show that there exists a Tychono? weakly Lindelöf space which does not have the property ${}^{\ast }{\mathcal{U}}_{fin}(\mathcal{O},\mathcal{O})$. This result answers the following open questions. [2] Does a weakly Lindelöf space have the property ${}^{\ast }{\mathcal{U}}_1(\mathcal{O},\mathcal{O})$? [3] Is there a Tychono? 1-star-Lindelöf space which does not have the property ${}^{\ast }{\mathcal{U}}_{fin}(\mathcal{O},\mathcal{O})$?