On a family of \(\lambda\)-topologies on a function space (Q1577167)

The author deals with the space \(C(X)\) of all continuous functions defined on a totally regular space \(X\). A set \(A\subset X\) is called bounded if every function in \(C(X)\) is bounded on \(A\). Each family \(\lambda\) of bounded subsets of \(X\) such that \(\widetilde\lambda =\cup\lambda\) is dense in \(X\) determines the so-called \(\lambda\)-topology on \(C(X)\) which is the topology of uniform convergence on the elements of \(\lambda\). The family \(\Lambda T(X)\) of all \(\lambda\)-topologies splits into the classes \(\Lambda_s T(X)=\{T_{\lambda}: \widetilde\lambda =s\}\) over dense subsets \(s\) of \(X\). Each family \(\mathcal A\subset \Lambda T(X)\) generates the topologies \(\inf\mathcal A\) and \(\sup\mathcal A\), the greatest lower and least upper bounds of \(\mathcal A\) in the complete lattice of all locally convex topologies on \(C(X)\). The main results are as follows: (1) the family \(\sigma\Lambda T(X)\) of all weak topologies generated by \(\Lambda T(X)\) does not coincide with \(\Lambda T(X)\) and the intersection of these families contains only the topologies of pointwise convergence; (2) the same holds for the family \(\inf\Lambda T(X)\); (3) \(\inf\Lambda_s T(X)=\Lambda_s T(X)\) if and only if every closed bounded subset of \(X\) is compact; (4) the operations \(\sigma\) and \(\inf\), as well as \(\sigma\) and \(\sup\), commute (\(\sigma\inf =\inf\sigma\) and \(\sigma\sup =\sup\sigma\)).