On the Graph Convergence of Sequences of Functions

Abstract

Let $(X,T_X)$ and $(Y,T_Y)$ be topological spaces. A sequence $(f_n)$ of functions $f_n:X \to Y$ is graph convergent to $f:X\to Y$ if for each set $U \in T_X\times T_Y$ containing the graph $Gr(f)$ of $f$ there is an index $k$ such that $Gr(f_n) \subset U$ for $n >> k$. It is proved that if $(X,T_X)$ is a $T_1$ space, then the graph convergence implies the pointwise convergence. Moreover the uniform and graph convergences are compared, and the graph limits of sequences of continuous (quasicontinuous, cliquish, almost continuous or Darboux) functions are investigated.