## A note of F-topologies.(English)Zbl 0712.54001

Let X be a set and let $$2^ X$$ denote its power set. A mapping $$u: 2^ X\to 2^ X$$ is called an F-topology on X if 1) $$u\phi =\phi$$; 2) $$A\subseteq uA$$; 3) $$A\subseteq B\Rightarrow uA\subseteq uB;$$ and 4) $$u(uA)=uA$$. Recall that any transitive binary relation on a set S is called a quasi-order on S. We denote by A(X) the set of all quasi-orders $$\rho$$ on $$2^ X$$ satisfying the additional conditions: i) $$B\subseteq A\Rightarrow A\rho B$$; ii) $$\phi \rho A\Rightarrow A=\phi$$; and iii) if $$A\in 2^ X$$ and $$(B_ i)_{i\in I}$$ is a family in $$2^ X$$ such that $$A\rho B_ i$$ for all $$i\in I$$, then $$A\rho\cup_{i\in I}B_ i$$. Now the main result of the paper under review can be stated as follows: Theorem. Let $${\mathcal B}$$ be a cover of X and let u be an F-topology on X. Then $${\mathcal B}$$ is an open base of u if and only if, for each pair of sets $$A,B\in 2^ X$$, there holds $B\subseteq uA\Leftrightarrow (\forall C)(C\in {\mathcal B}\text{ and } A\subseteq X\setminus C\Leftrightarrow B\subseteq X\setminus C).$ Corollary. Let $$\rho$$ be a binary relation on $$2^ X$$. Then $$u\in A(X)$$ if and only if there exists a cover $${\mathcal B}$$ of X such that, for each pair of sets $$A,B\in 2^ X$$, there holds $A\rho B\Leftrightarrow (\forall C)(C\in {\mathcal B}\text{ and } A\subseteq X\setminus C\Rightarrow B\subseteq X\setminus C).$
Reviewer: P.Morales

### MSC:

 54A05 Topological spaces and generalizations (closure spaces, etc.) 06A99 Ordered sets

### Keywords:

F-topology; quasi-order
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### References:

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