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