One-point compactifications of ditopological texture spaces

Ditopological texture spaces and fuzzy topology, I. Basic concepts

“…However, in place of the full lattice of subsets of some base set S, attention is now focused on a suitable subfamily of subsets, called a texturing of S, and within this context bitopologies are replaced by dichotomous topologies, or ditopologies for short. Some results on ditopological texture spaces may be found in [3,4,7,8,14,16]. Our aim in this series of papers is to place the investigation of textures and ditopological texture spaces on a ÿrm categorical footing, and in particular to use the representations of fuzzy sets as texture spaces discussed in [5,6] to study the relation between L-topologies and ditopological texture spaces in greater detail.…”

“…We recall [2][3][4][5][6][7][8]14,16] that a texturing on S is a point separating, complete, completely distributive lattice S of subsets of S with respect to inclusion, which contains S, ∅, and for which meet coincides with intersection and ÿnite joins ∨ coincide with unions ∪. The pair (S; S) is then known as a texture.…”

Ditopological texture spaces and fuzzy topology, II. Topological considerations

“…Then if and Ä are related by Ä = [ ] we say that ( ; Ä) is a complemented ditopology on (S; S; ) [5,6,12,13,20]. In this case we have (…”

“…Otherwise the series is largely self-contained, although the reader may wish to refer to [4][5][6][7][8][12][13][14][15][16]20,28] for additional background and motivation. The reader is referred to [17] for terms from lattice theory not deÿned here.…”

A Wallman-type compactification of texture spaces

“…It is known that the category cdfSDitop is equivalent to H where H is the category of Hutton spaces and the morphisms : [7,Proposition 5.6]. Therefore, if (L, L ) is a Hutton space, then (2 × L, 2×L ) is a dicompact extension of (L, L ) [11]. Hence, ditopological texture spaces give useful ways of looking at certain lattice theoretic aspects of topology and variable basis topology [16].…”