Some Categorical Aspects of Fuzzy Topology

Abstract

Research in recent years has revealed that the construct of fuzzy topological spaces behaves quite differently from that of topological spaces with respect to certain categorical properties. In this paper we discuss some of these aspects. Since the topological construct L-FTS contains nontrivial both initially and finally closed full subconstructs, and each such construct gives rise to a natural autonomous theory of fuzzy topology, it can be said to some extent that fuzzy topology should consist of a system of closely related topology theories, including the classical topology theory as a special case, with each applying to one such subconstruct. Therefore in the first part of this paper the theory of sobriety is established for each finally and initially closed full subconstruct of L-FTS to illustrate this idea. The second topic of this paper is the relationship between the construct of stratified L-fuzzy topological spaces and several other familiar constructs in fuzzy topology, for example, the constructs ofSostak fuzzy topological spaces and L-fuzzifying topological spaces. © 2000 Academic PressKey Words: fuzzy topology; sobriety; frame; topological construct; co-tower extension; total fuzzy topology. PRELIMINARIESAfter a development period of more than 30 years, fuzzy topology has become rather diverse in its topics as well as its methods. The recent book [17] is a comprehensive treatment of the general topology and lattice theoretic aspects of fuzzy topology. In this paper we deal with some categorical properties of the construct of fuzzy topological spaces.

Strong fuzzy compact sets and ultra-fuzzy compact sets in L-topological spaces

“…We use N to denote the set of all natural numbers, Y the characteristic function of Y (sometimes we make no distinction between Y and Y ), |Y | the cardinal number of Y , [ ] ∈ L X the L-subset of X taking constant value , suppA the support of A, A |Y the restriction of A to Y , and write x = [ ] ∧ {x} , where is called the height of x (∅ = Y ⊂ X; ∈ L; ∈ L − {0}; A ∈ L X ; x ∈ X ). Analogous to [10], we call A ∈ L X an n-step L-subset (where n is a cardinal number) i |{A(x) | x ∈ X } − {0}| = n; the largest value (if it exists) of an n-step L-subset is called its height. Obviously, every crisp L-subset is 1-step.…”

Sobriety and Localic Compactness in Categories ofL-Bitopological Spaces

“…These constructions allow the replacement of SLoc if L is a frame. Also, [2,7,8] yield a class of adjunctions and equivalences indexed by L ∈ SFrm which set up classes of Stone representation theorems and Stone-Čech compactifications with appropriate restrictions on L. Finally, many of the ideas concerning the class of basic adjunctions and equivalences were anticipated by Höhle [9].…”