QN-spaces, wQN-spaces and covering properties
Let \(X\) be an infinite Hausdorff space. \(X\) is an \({\mathcal S}^*\)-space if \({\mathrm{scl}} ({\mathrm{scl}} (A))={\mathrm{scl}}(A)\) for every subset \(A \subseteq X\) where \({\mathrm{scl}} (A)\) denotes the sequential closure of \(A\). \(X\) satisfies \((\alpha_1)\) if for any \(x \in X\) and any collection \(\{(x_{n,m})_m : n \in {\mathbb{N}} \}\) of sequences converging to \(x\), there is a sequence \((x_n)_n\) converging to \(x\) which contains all but finitely many members of every \((x_{n,m})_m\). If we only require that \((x_n)_n\) contains a member of \((x_{n,m})_m\) for infinitely many \(n\), then \(X\) is said to satisfy \((\alpha_4)\). If we can always find an unbounded non-decreasing sequence of natural numbers \((n_m)_m\) such that \((x_{n_m,m})_m\) converges to \(x\), then we say \(X\) satisfies \((\alpha_0)\). Properties \((\alpha_1)\) and \((\alpha_4)\) were introduced by Arkhangel'skiǐ and \((\alpha_0)\) by the authors. Clearly, \((\alpha_1)\) implies \((\alpha_0)\) which in turn implies \((\alpha_4)\). A sequence \((f_n)_n\) of real-valued functions on \(X\) converges quasinormally to a function \(f\) if there is a sequence \((\epsilon_n)_n\) of positive reals converging to \(0\) such that for all \(x \in X\), \(| f_n (x) - f(x) | < \epsilon_n\) holds for all but finitely many \(n\). Quasinormal convergence is stronger than pointwise convergence and weaker than uniform convergence. \(X\) is a QN-space if every sequence \((f_n)_n\) converging pointwise to \(0\) also converges quasinormally to \(0\). \(X\) is a wQN-space if every such \((f_n)_n\) contains a subsequence converging quasinormally to \(0\). \(C_p (X)\) denotes the space of real-valued functions on \(X\) with the topology of pointwise convergence. An infinite cover \({\mathcal U}\) of \(X\) is a \(\gamma\)-cover if every \(x \in X\) belongs to all but finitely many members of \({\mathcal U}\). A \(\gamma\)-cover \({\mathcal U}\) is shrinkable if there is a closed \(\gamma\)-cover which refines \({\mathcal U}\). The covering property \({\mathrm{S}}_1 (\Gamma_{{\mathrm{shr}}},\Gamma)\) says that for every sequence \(({\mathcal U}_n)_n\) of open shrinkable \(\gamma\)-covers there exists \(U_n \in {\mathcal U}_n\) such that \(\{ U_n : n \in \mathbb{N} \}\) is a \(\gamma\)-cover. It is well-known that \(C_p (X)\) is an \({\mathcal S}^*\)-space iff it has property \((\alpha_4)\) iff \(X\) is a wQN-space. The authors prove that, for a normal space \(X\), any of these properties is equivalent to \({\mathrm{S}}_1 (\Gamma_{{\mathrm{shr}}},\Gamma)\). They also show that, for a Hausdorff space \(X\), \(C_p (X)\) has property \((\alpha_1)\) iff it has property \((\alpha_0)\) iff \(X\) is a QN-space. For perfectly normal spaces \(X\), the latter is also characterized in terms of covering properties.
scientific article; zbMATH DE number 5131636
scientific article