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.

中文翻译：

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泛函 Alexandroff 拓扑的格子

如果 f : X → X $f:X \rightarrow X$ 是一个函数，则 X 上关联的泛函 Alexandroff 拓扑是拓扑 P f，其闭集为 { A ⊆ X : f ( A ) ⊆ A } $\{A \subseteq X : f(A) \subseteq A\}$ 。我们提出了有限集 X 上泛函 Alexandroff 拓扑的特征，并表明有限集 X 上所有泛函 Alexandroff 拓扑的集合 FA ( X )，按包含排序，是一个补格。