Several Fuzzy Approximation Spaces And Their Topological Structures

Fuzzy set theory and rough set theory are both the mathematical tools for dealingwith uncertainties. Fuzzy rough set or fuzzy approximation space is the generalizationof fuzzy set and rough set, which has been one of important research. Under some view,lower and upper approximation operators in approximation spaces have the similar prop-erties with the interior and closure operators in topological spaces. Thus approximationoperators draw close links between approximation spaces and topological spaces, whichillustrates that we can study fuzzy approximation spaces by means of topology, and viceversa. This thesis is devoted to the discussion of several class of fuzzy approximation s-paces and their topological structures, and study several class of fuzzy relations by meansof topology. We show The fact that there exists a one-to-one correspondence between theset of some fuzzy relations and the set of some fuzzy topologies.This thesis is composed of five chapters. In Chapter1, we give a survey of the back-grounds and modern developments of fuzzy rough sets and their topological structures,and describe the main research content of this thesis. In Chapter2, we introduce someelementary concepts and results of the fuzzy sets and topological theory which will beused in this thesis.In Chapter3, fuzzy rough approximations are further investigated and topologicalstructures of fuzzy approximation spaces are studied. we get the following results:(1) Suppose that τ is a Alexandrov fuzzy topology on U and Rτis the fuzzy relationinduced by τ on U. Let τRτbe the fuzzy topology induced by Rτon U. Then τRτ=τ if and only if τ satisfies (CC) axiom.(2) Let Σ={R: R is a preorder fuzzy relation on U},Γ={τ: τ is a Alexandrovfuzzy topology on U satisfying (CC) axiom}. Then there exists a one-to-one correspon-dence between Σ and Γ.In Chapter4, we generalize fuzzy set environments to interval-valued fuzzy set envi-ronments and study the interval-valued fuzzy approximation spaces and their topologicalstructures, and obtain sufcient conditions that every interval-valued fuzzy topologicalspace is an interval-valued fuzzy approximating space. The main result is the following:Assume that τ is an interval-valued fuzzy topology on U. Let Rτbe the interval-valuedfuzzy relation induced by (U, τ) and τRτbe the interval-valued fuzzy topology inducedby Rτon U. If τ satisfies (C1) and (C2) axioms, then τRτ=τ.In Chapter5, we generalize fuzzy set environments to L-fuzzy set environments, study the L-fuzzy soft approximation spaces and their topological structures, and get thefollowing results:(1) Assume that τ is a L-fuzzy topology on X and (fτ)Eis the L-fuzzy soft setinduced by τ on X. Let τfτbe the L-fuzzy topology induced by (fτ)Eon X. Thenτ=τfτ.(2) Let Σ={fE: fEis an L-fuzzy soft set over X} and Γ={ρ: ρ is an L-fuzzy relationfrom E to X}. Then there exists a one-to-one correspondence between Σ and Γ.