Discrete reflexivity in topological groups and function spaces

We show that pseudocharacter turns out to be discretely reflexivein Lindelöf \(\Sigma\) $\mathrm{\Sigma }$ -groups but countable tightness is notdiscretely reflexive in hereditarily Lindelöf spaces. We alsoestablish that it is independent of ZFC whether countablecharacter, countable weight or countable network weight isdiscretely reflexive in spaces \(C_p(X)\) $C_p(X)$ . Furthermore, we provethat any hereditary topological property is discretely reflexivein spaces \(C_p(X)\) $C_p(X)$ with the Lindelöf \(\Sigma\) $\mathrm{\Sigma }$ -property. If \(C_p(X)\) $C_p(X)$ is a Lindelöf \(\Sigma\) $\mathrm{\Sigma }$ -space and \(L D\) $\mathit{LD}$ is a \(k\) $k$ -space for any discrete subspace \( { D C_p(X) } \) $D{C}_p(X)$ , then it isconsistent with ZFC that \(C_p(X)\) $C_p(X)$ has the Fréchet–Urysohnproperty. Our results solve two published open questions.