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On a generalization of approximation space. (English) Zbl 0755.04011

Summary: We consider approximation spaces as ordered triples \((U,F,G)\) composed of a nonempty set \(U\) and two set-theoretical operations \(F\), \(G: P(U)\to P(U)\) such that for any \(X\subseteq U: F(X)=\bigcup\{F(\{x\}): x\in X\}\) and \(G(x)=\{y\in U:\emptyset\neq F(\{y\})\subseteq X\}\). Operations \(F\), \(G\) are in particular, respectively, the upper and lower approximation operations of Z. Pawlak and certain variants of the latter. We establish various properties of \(F\), \(G\) and at the same time known and new properties of approximations. In particular, we give necessary and sufficient conditions for \(F\), \(G\) to be the topological closure and interior operations in the sense of K. Kuratowski [Fundam. Math. 3, 182-199 (1922; JFM 48.0210.04)], respectively.

MSC:

03E99 Set theory
54A99 Generalities in topology
03E20 Other classical set theory (including functions, relations, and set algebra)

Citations:

JFM 48.0210.04
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