On extension of maps in topological spaces (Q1114954)

In this paper strongly nbd-finite families of subsets of a topological space are introduced and investigated and then used to investigate piecewise definitions of irresolute maps, semi-continuous maps, and semihomeomorphisms. A family \(\{A_ m|\) \(m\in M\}\) of subsets of a space X is strongly nbd-finite if for each \(x\in X\), there is an open set V containing x, satisfying one of the following conditions: (a) \(V\cap A_ m=\emptyset\) for every \(m\in M\), (b) there is a nonempty finite subset N of M such that i) \(V\cap A_ m\neq \emptyset\) for every \(m\in N\), ii) \(V\cap A_ m\subset A_ k\) for every \(m,k\in N\), and iii) \(V\cap A_ m=\emptyset\) for every \(m\not\in N\).