Uniform boundedness of a family of weakly regular nonadditive mappings (Q2266078)

The following result is proved. Let (T,\({\mathcal T})\) be a regular Hausdorff space, let Q denote the set of positive rationals. Let \({\mathcal B}=\{B_ p\}_{p\in Q}\) be a family of neighbourhoods in a topological Abelian semigroup X satisfying \(B_ p\subset B_ q\) whenever \(p<q\), and \(B_ p+B_ q\subset B_{p+q}.\) Let R be a ring and let K be a set of multi- functions \(\Phi\) : \(R\to P(X)\) which are weakly \({\mathcal B}\)-regular [that is, for each \(p\in Q\) and \(E\in R\) there is a compact \(K\in R\cap E\) with \(\Phi (E\setminus K)\subset B_ p]\) and satisfy \(\Phi (E)\subset B_ p,\quad \Phi (F)\subset B_ q\) imply \(\Phi (E\cup F)\subset B_{(p+q)k}\) and \(\Phi (E\cup F)\subset B_ p,\quad \Phi (E)\subset B_ q\) imply \(\Phi (F)\subset B_{(p+q)k}.\) Then K(R) is \({\mathcal B}\)-bounded if and only if K is locally bounded on \({\mathcal T}\), where the two boundedness notions are defined in the paper in a natural way.