Spaces with large weak extent (Q409523)

The weak extent \(we(X)\) of a space \(X\) is the smallest cardinal number \(\kappa\) such that for each open cover \(\mathcal U\) of \(X\) there is a subset \(A\) of \(X\) such that \(|A| \leq \kappa\) and \(St(A,\mathcal U) = X\). In 1998, \textit{M. V. Matveev} asked if there is a Tychonoff centered-Lindelöf space \(X\) with \(we(X)>\mathfrak c\). The author shows that for any cardinal \(\kappa\) there is a Tychonoff pseudocompact centered-Lindelöf space \(X\) with \(we(X)\geq k\); under \(2^{\aleph_0}= 2^{\aleph_1}\) there is a normal centered-Lindelöf space \(Y\) such that \(we(Y) \geq \omega_1\).