Exponential sections (Q579660)

The author considers mappings \(F: Y\to 2^ X\), where \(2^ X\) is the family of nonempty closed subsets of X in the Vietoris topology, and such maps restricted to a subspace P(X) of \(2^ X\). A map \(f: Y\to X\) is a section of F if f(y)\(\in F(y)\) for every \(y\in Y\), and P(X) has a section of class \(\alpha\) if the identity \(1_{P(X)}\) has a section of class \(\alpha\). A space X is semicircular if for every \(x\in X\) there exists a sequence of open neighborhoods \(U_{nx}\), \(n\in N\), such that 1) \(\cap_{n\in N}U_{nx}=\{x\}\), 2) if \(\{x_ n\}\) is any sequence such that \(x\in U_{nx_ n}\) for all n then \(x_ n\to x\). CML(X) is the family of nonempty complete subsets of a semicircular space X. The author proves that if X is a null-dimensional paracompact semicircular space, then CML(X) has a continuous section, and that if X is a regular complete semicircular space then \(2^ X\) has a section of the first class. These generalize results of \textit{M. Choban} [Tr. Mosk. Mat. O.-va 22, 229-250 (1970; Zbl 0231.54013)].