The Lattice of Functional Alexandroff Topologies

If f : X → X $f:X \rightarrow X$ is a function, the associated functional Alexandroff topology on X is the topology P f whose closed sets are { A ⊆ X : f ( A ) ⊆ A } $\{A \subseteq X : f(A) \subseteq A\}$ . We present a characterization of functional Alexandroff topologies on a finite set X and show that the collection F A ( X ) of all functional Alexandroff topologies on a finite set X , ordered by inclusion, is a complemented lattice.

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