Discrete reflexivity in topological groups and function spaces

Abstract

We show that pseudocharacter turns out to be discretely reflexive in Lindelöf \(\Sigma\)-groups but countable tightness is not discretely reflexive in hereditarily Lindelöf spaces. We also establish that it is independent of ZFC whether countable character, countable weight or countable network weight is discretely reflexive in spaces \(C_p(X)\). Furthermore, we prove that any hereditary topological property is discretely reflexive in spaces \(C_p(X)\) with the Lindelöf \(\Sigma\)-property. If \(C_p(X)\) is a Lindelöf \(\Sigma\)-space and \(L D\) is a \(k\)-space for any discrete subspace \( { D C_p(X) } \), then it is consistent with ZFC that \(C_p(X)\) has the Fréchet–Urysohn property. Our results solve two published open questions.