Graphical convergence of continuous functions (Q2448973)

If \(X\) and \(Y\) are topological spaces and \(C(X,Y)\) is the set of all continuous mappings from \(X\) into \(Y\), then the graph topology on \(C(X,Y)\) has basic sets of the form \(V^+ = \{f\in C(X,Y):\Gamma(f) \subset V\}\), \(V\) is open in \(X\times Y\), where \(\Gamma(f)\) is the graph of \(f\). It is shown that if \(X\) is a \(T_1\) space and \(Y\) is a topological space, then the graph topology, the finite topology and the locally finite topology on \(C(X,Y)\) coincide. The main result states: if \(X\) and \(Y\) are metric spaces, then a net \((f_\lambda)\) in \(C(X,Y)\) converges to a function \(f\) in the graph topology if and only if for every metric \(\rho\) on \(X \times Y\) compatible with the product topology the net \((\rho(\cdot,\Gamma(f_\lambda))\) of the corresponding distance functionals uniformly converges to \(\rho(\cdot, \Gamma(f))\).