Determination Of Soft Topologies And Finite Matroids

Order isomorphism is an important concept in mathematics. For a given set X, let T(X) be the set of all topology on X, CL(X) be the set of all Kuratovski closure operator on X. If we can give the partial order relations<on CL(X) and the isomorphism F:(CL(X),≤)→(T(X),∈), then we can say that we can use Kuratovski closure operator to determine the topology. This paper studied determination of soft topologies and finite matroids.The construction and the main contents of this paper are as follow:Chapter1Preliminaries. In this chapter,we gave the basic concepts and rusults of soft set,soft topology and matroid.Chapter2The determination of soft topologies. In this chapter, we studied cryptomorphic properties of soft topologies. For an arbitrary set X, appropriate or-der relations on SWCL(X.1)(the set of all soft weak closure operators indexed by1), SW1N(X,I)(the set of all soft weak interior operators indexed by I), SWOU(X,I)(the set of all soft weak exterior operators indexed by I) and SWB(X, I)(the set of all soft weak boundary operators indexed by1) can be defined respectively, which make SWCL(X,I), SWIN(A,I), SWOU(X,I) andB(X,I) to be complete lat-tices that are isomorphic to (ST(X,I),∈), where ST(X,I) is the set of all soft topologies on X indexed by I.Chapter3The determination of matroids. In this chapter, we proved that: for an finite set E, appropriate order relations on S(E)(the set of all systems of support sets on E), SC(E)(the set of all systems of non-support sets on E), and H(E)(the set of all systems of hyperplane sets on E) can be defined respectively, which make S(E),SC(E) and H(E) to be isomorphic to (B(E),≤), where B(E) is the set of all systems of base sets on E.