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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
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References:

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