On the Reflectiveness and Coreflectiveness of Subcategories of FTS

Abstract

PreliminariesWe assume familiarity with most of the usual notational conventions in the field and limit ourselves to recalling only the most important concepts and definitions required in this paper.Our basic category is FTS which stands for the topological category [l] of fuzzy topological spaces and continous maps as introduced in [3]. MAX stands for the full isomorphism closed subcategory of l?TS which was essentially introduced in [9]. In order to reformulate its definition we recall the notion of level topologies introduced in [5]. J i (X, A ) E lFTSl and a E I, then the a-level topology of A is defined as r.(d) := {p-I(]a, 11) I p E A } . Then by definition (X, A ) E (MAX( if and only if for any other fuzzy topology r on X with level topologies identical to thosa of d, i.e. such that &.(A) = c,(P) for all a E I,, it follows that I'c d.If no confusion can occur, the topological space (X, &.(A)) is often shortly denoted FNS stands for the full isomorphism closed subcategory of FTS coneisting of those fuzzy topological spaces which are generated by a fuzzy neighborhood system. This category waa introduced in [6], and the objects are also called fuzzy neighborhood spaces. We recall that a fuzzy neighborhood system on a set X consists of a collection of prefilters ( J~( z ) ) , , ,~where -denotes the so-called saturation operation defined by b:= (4 I v E E I,, 3 4, E S:(#l, -E 5 4}.The family ( Y , ) , ,~ in (FN3) is called an €-kernel for Y.

A note on fuzzy neighbourhood base spaces

“…It was shown that MAX [10] is a bicoreflective subcategory of SI-TOP but it is not a reflective subcategory of SI-TOP, that is, MAX is not stable.…”

“…We write c ( ) for ( ( )) c , the set of closed subsets of (X, ( )). In [14] the full subcategory MAX of SI-TOP was essentially introduced, then its reformulation, in [10], was defined in the following way: the relation between objects (X, ) and (X, ) of SI-TOP, defined by…”

Basic structures of fuzzy topology

“…(A fuzzy topological space is called weakly induced if all u E 7" are lower semicontinuous when considered as mappings u : (X, ~-N 2 x) -> I, [135].) The category w(Top) of all topologically generated spaces is coreflexive in LeFT(I), the coreflexiones given by Lw: (X,T) ---> (X,~wT) [126].…”

Applications of Category Theory to Fuzzy Subsets

“…One of their nice aspects is that they can also be characterized as topological categories in which coproducts and quotients are preserved under pullbacks along projections. UNIF) as bicorefiective subcategories closed under the formation of finite products [Wuyts, Lowen and Lowen 1988;Lowen, Wuyts and Lowen 1989]. The categories FTS and FNS (resp.…”