Topologies on function spaces (Q5917538)

A map \(f\) of a space \(Y\) into a space \(Z\) is called clopen continuous at \(y\in Y\) if for every open neighborhood \(V\) of \(f(y)\) there exists an open and closed neighborhood \(U\) of \(y\) such that \(f(U)\subset V\). The map is clopen continuous on \(Y\) if it is clopen continuous at each point of \(Y\). The set of all clopen continuous maps of \(Y\) into \(Z\) is denoted by \(COC(Y, Z)\). In this paper the author discusses some convergence properties of the function space \(COC(Y, Z)\), and obtains some sufficient and necessary conditions for clopen splitting topology or clopen joining topology on \(COC(Y, Z)\).