@MISC{Bancerek_mmlidentifier:, author = {Grzegorz Bancerek}, title = {MML Identifier: WAYBEL19.}, year = {} }

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Abstract

Let T be a non empty FR-structure. We say that T is lower if and only if: (Def. 1) {(↑x) c: x ranges over elements of T} is a prebasis of T. Let us note that every non empty reflexive topological space-like FR-structure which is trivial is also lower. Let us note that there exists a top-lattice which is lower, trivial, complete, and strict. One can prove the following proposition (1) For every non empty relational structure L1 holds there exists a strict correct topological augmentation of L1 which is lower. Let R be a non empty relational structure. Note that there exists a strict correct topological augmentation of R which is lower. We now state the proposition (2) Let L2, L3 be topological space-like lower non empty FR-structures. Suppose the relational structure of L2 = the relational structure of L3. Then the topology of L2 = the topology of L3. Let R be a non empty relational structure. The functor ω(R) yielding a family of subsets of R is defined by: (Def. 2) For every lower correct topological augmentation T of R holds ω(R) = the topology of T. One can prove the following propositions: (3) Let R1, R2 be non empty relational structures. Suppose the relational structure of R1 = the relational structure of R2. Then ω(R1) = ω(R2). (4) For every lower non empty FR-structure T and for every point x of T holds (↑x) c is open and ↑x is closed.