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

A weakly Lindel\"of space which does not have the property $^*\mathcal{U}_{fin}(\mathcal{O},\mathcal{O})$