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

TL;DR

The paper constructs a Tychonoff weakly Lindelöf space lacking a specific selection property, answering open questions about the relationship between weakly Lindelöf spaces and certain covering properties.

Contribution

It provides a counterexample of a weakly Lindelöf space without the property $^*\mathcal{U}_{fin}( ext{O}, ext{O})$, addressing open problems in topology.

Findings

Existence of a Tychonoff weakly Lindelöf space without $^*\mathcal{U}_{fin}( ext{O}, ext{O})$

Answers open questions about properties of weakly Lindelöf spaces

Clarifies the relationship between weakly Lindelöf and certain selection properties

Abstract

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 $^*\mathcal{U}_{fin}(\mathcal{O},\mathcal{O})$?