Closed graph property in Alexandroff spaces

Abstract

In the following text we show if $X$ is an Alexandroff space, then $f:X\to Y$ has closed graph if and only if it has constant closed value on each connected component of $X$. Moreover, if $X$ an Alexandroff space and $f:X\to Y$ has closed graph, then $f:X\to Y$ is continuous. As a matter of fact, the number of maps which have closed graph from Alexandroff space $X$ to a topological space $Y$ depends just on the the number of connected components of $X$ and the number of closed points of $Y$.