Closed graph property in Alexandroff spaces

TL;DR

The paper characterizes maps with closed graphs on Alexandroff spaces by showing they are precisely those that are constant on each connected component and take values in closed points of the target space. This leads to an exact counting formula for such maps: the number equals $\\beta^\\alpha$, with $\\alpha$ the number of components of $X$ and $\\beta$ the number of closed points of $Y$, and it implies these maps are automatically continuous. The Khalimsky space example illustrates the structure of closed-graph maps in connected settings and highlights that continuous constants need not have closed graphs. Overall, the work connects component structure with closed-graph behavior in a combinatorial/topological framework, yielding both structural insight and counting consequences.

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 $Y$.

Theorems & Definitions (15)

• Lemma 1
• proof
• Theorem 2
• proof
• Remark 3
• Theorem 4
• proof
• Lemma 5
• proof
• Lemma 6