Paracompactness and Full Normality in Ditopological Texture Spaces

Abstract

In this paper the authors lay the foundation for a theory of dicovers of ditopological texture spaces and use this to define notions of paracompactness and full normality. Some applications to fuzzy topology are also mentioned. ᮊ 1998 Academic PressKey Words: texture space; ditopology; dicover; dicover paracompact; dicover co-paracompact; dicover biparacompact; dicover fully normal; fuzzy topology; ␣-paracompact; ␣ , ⑀-paracompact

Di-uniformities and Hutton uniformities

“…General references on ditopological texture spaces include [1,2,3,4,5,6] and constant reference will be made to [13,14,15,16,17] for definitions and results relating to di-uniformities. Our standart references for quasi uniformity are [7,8,9].…”

“…We recall from [2] that by a difamily we mean a set C = {(A j , B j ) | j ∈ J} of elements of S × S and C is called a dicover of (S, S) if j∈J1 B j ⊆ j∈J2 A j for all partitions (J 1 , J 2 ) of J. A dicover corresponds to a dual cover in the sense of [1] and this notion is related to the notion of pairs of covers with a common index used by Gantner and Steinlage [8] to characterize quasi uniformities.…”

Di-uniform texture spaces

“…The reader is referred to [2,3,6,7] for some results on ditopological texture spaces and their relation with fuzzy topologies. A subset β of τ is called a base of τ if every set in τ can be written as a join of sets in β, while a subset β of κ is a base of κ if every set in κ can be written as an intersection of sets in β.…”

“…One such is that of direlation and difunction, another that of dicover ( [2,3], see §2 below). Both play a crucial role in this paper.…”

“…A second important formulation of the theory of uniformities is that of the covering uniformity [12], so we will require an appropriate notion of cover in order to obtain an analogous description for textures. In this section we show that the notion of dicover, used in [2] to characterize the important form of compactness mentioned above and in [3] to describe various covering properties of ditopological texture spaces, is associated in a natural way with direlations. Hence this notion will form the basis for our description of covering uniformities in the textural sense.…”