Research On Some Properties Of The Relative Topology

Abstract:In the research background of general topological properties and relative topological properties, relative properties presented by AV Arhangel skii have be surveyed from several aspects which include relative T2 space, relative T3 space, relative regular space and relative normal space. Then we draw a conclusion that any Y, which is a subset of X, is relative normal in X, and so on. Meanwhile, the sub-regularity of the relative topology is also given.The definition of the weak irreducible element is given on the basis of irreducible element. Meanwhile we promote the L—topological spaces to the L—pre-topological spaces. On the basis of the separation of the L—topological spaces, we introduce the sub-separation of the L—pre-topological spaces including sub-T1, sub-T2, sub-T3, sub-T4, and proving that if the topological system (copr(Lx), δ,|=) which is induced by the L—pre-topological spaces is sub-T1, then the L—pre-topological spaces is sub-T1.The main contents of this paper are as follows:The first part:The concepts of the relative topology, the L—topological spaces, the lattice theory and topological system are given.The second part:The sub-separation of L—pre-topological spaces are introduced, and the relations between them are given too. Then the relative separation and rela-tive sub-separation of the L—Fuzzy interior spaces are defined, and their equivalent characterizations are given. And the relative separation which is heritable is given too.The third part:Some properties of the sub-topological systems are given in our research, based on the definition of sub-topological systems. In addition, then the relative separation and relative compactness of the sub-topological systems are given, and then the definitions of sub-topological systems named Ti(i= 1,2,2^,3,4) are given on this basis too. In addition, it is also proved that the sub-topological system E is relative to the topological-system D are Ti if and only if the Spat(E) is relative to Spat(D) are Ti. And the sub-topological system E is relative to the topological-system D is compact if and only if the Spat(E) is relative to Spat(D) is compact.