Determinination Of L-fuzzy Topologies And Connectivity Of Weak Topological Molecular Lattices

Determining a topology is an interesting question. For a given set X,let T(X)be the set of all topologies on X,and CL(X) the set of all Kuratovski closure operators on X.If one can give both an appropriate order relation≤on CL(X) and an order isomorphism from (CL(X),≤)to (T(X),(?)),then we say that topologies and closure operators can be determined reciprocally. It can be proved that topologies and each of closure operators, interior operators,exterior operators, boundary operators, derived operators, difference derived operators,neighborhood operators, remote neighborhood operators,and convergence classes of nets can be determined reciprocally. Similar results on L-fuzzy topologies will be proved in this paper.The main points of this paper are as follows:In the first chapter,some concepts on L-fuzzy topologies, L-fuzzy interior oper-ators,L-fuzzy neighborhood operators,L-fuzzy closure operators,weak topological molecular lattices, and category, and some important conclusions are introduced.Let X be a set,L a Hutton algebra, FT(X,L) (resp.,FN(X,L),FI(X,L), FC(X, L))denote the set of all L-fuzzy topologies on X (resp.,the set of all L-fuzzy neighborhood operators, the set of all L-fuzzy interior operators,the set of all L-fuzzy closure operators).The second chapter gives a one-to-one correspondenceψ32(resp.,ψ34) from FI(X, L)to FN(X, L)(resp.,to FC(X,L))and a one-to-one correspondenceψ24 from FN(X, L) to FC(X, L).It is proved that an appropriate order relation may be defined on FT(X, L) (resp.,FN(X, L),FI(X, L),FC(X, L)) such thatψ32,ψ34 andψ24 are all complete lattice isomorphisms.In the third chapter, the notion of connected element of weak topological molec-ular lattices is defined and their basic properties (including productivity) are dis-cussed. Moreover,local connectedness of weak topological molecular lattices is also studied.Let PordField be the category of partial ordered fields and mappings which both preserves orders and preserves operations,Field be the category of fields and mappings which preserves operations, L-FCCS be the category of L-fuzzy closure systems spaces and continuous mappings,and Set be the category of sets and map-pings, In the fourth chapter,it is proved that PordField is topological category on Field but not a topological construct on Set,and L-FCCS is a topological construct on Set.