Continuity phenomenon of Kenderov and porosity: the case of countable systems (Q6620705)

For multifunctions acting from a topological space \(X\) into a set \(Y\) (without topological structure) there is a substitute of lower semicontinuity called \textit{(\(A\)-)lower almost continuity} (briefly \textit{lac}) and investigated by \textit{P. S. Kenderov} [Serdica 9, 149--160 (1983; Zbl 0539.54010)]. Each multifunction \(F: X \rightarrow Y\) has this property except a set of first category in \(X\). The question arises, can the \(\sigma\)-ideal of first category sets be repaced by some smaller \(\sigma\)-ideal?\N\NIn case when \(X\) is a metric space, a natural candidate is the \(\sigma\)-ideal of \(\sigma\)-porous sets (in case of finite dimensional Euclidean space it is a proper subideal of the \(\sigma\)-ideal of sets of first category and simultaneously of L-measure zero).\N\NIn the paper under review it is exhibited, that for \(X\) metric and \(Y\) arbitrary (without structure) \(F\) is (\(A\)-)lac except some \(\sigma\)-porous subset of \(X\). If \(X\) has also a linear structure (e.g. is a Banach space) the notion of cone porosity and \(A\)-cone lac are introduced and a variant of the above statement for \(A\)-cone lac \(F\) is proved.\N\NThen applications to submonotone mappings and sets of singlevaluedness are presented and corollaries concerning interesting particular cases (as Stegall's variational principles, monotone operators, minimization problems etc.) are indicated.