Real Dicompactifications of Ditopological Texture Spaces

Abstract

Real dicompactifications and dicompactifications of a ditopological texture space are defined and studied. Section 2 considers nearly plain extensions of a ditopological texture space (S, S, iota, kappa). Spaces that possess a nearly plain extension are shown to have a property, called here almost plainness, that is weaker than that of near plainness, but which shares with near plainness the existence of an associated plain space (S-p, S-p, tau(p), kappa(p)). Some properties of the class of almost plain ditopological texture spaces are established, a notion of canonical nearly plain extension of an almost plain ditopological texture space, projective and injective pre-orderings and the concept of isomorphism on such canonical nearly plain extensions are defined. In Section 3 the notion of nearly plain extension is specialized to that of real dicompactification and dicompactification, and the spaces that have such extensions are characterized. Working in terms of a specific representation of the canonical real dicompactifications and dicompactifications of a completely biregular bi-T-2 almost plain ditopological space, the interrelation between sub-T-lattices of the T-lattice of omega-preserving bicontinuous real mappings on the associated plain space and the real dicompactifications and dicompactifications are investigated. In particular generalizations of the Hewitt realcompactification and Stone-Cech compactification are obtained, and shown to be reflectors for the appropriate categories. (C) 2009 Elsevier B.V. All rights reserved.