Abstract
A function
$f:X\rightarrow X$
determines a topology
$P(f)$
on
$X$
by taking the closed sets to be those sets
$A\subseteq X$
with
$f(A)\subseteq A$
. The topological space
$(X,P(f))$
is called a functionally Alexandroff space. We completely characterise the homogeneous functionally Alexandroff spaces.