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, τ , κ). 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 , τ p , κ 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 biT 2 almost plain ditopological space, the interrelation between sub-T-lattices of the T-lattice of ω-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-Čech compactification are obtained, and shown to be reflectors for the appropriate categories.

Dicompleteness and real dicompactness of ditopological texture spaces

“…This paper continues the study of dicompact and real dicompact completely biregular bi-T 2 spaces begun in [24][25][26], and in particular relates this work with the completeness of certain compatible di-uniformities. Our source of inspiration is again [2,Chapter 3] on bitopological notions of compactness and real compactness, and other papers in this area such as [13].…”

“…We begin by extending the notion of nearly plain extension of almost plain ditopological texture spaces, discussed in [26], to nearly plain dicomplete extensions.…”

“…Unlike the classical case [15], the family BDF(S) of real bicontinuous difunctions [12,24], and the family BA(S) of real ωpreserving bicontinuous point functions [24], are not rings but T -lattices [2]. Such parts of the theory of T -lattices developed in [2] as are required here have already been given in [24][25][26], and will not be repeated. A theory of di-uniformities on textures has been developed in [18], and the relationship with classical uniformities and quasi-uniformities discussed in [19].…”

“…Since our main aim in considering completeness is to link this with our earlier work on real dicompactness, we continue to work within the same framework. Hence, in Section 2, we begin by extending the notion of a nearly plain extension of an almost plain ditopological texture space introduced in [26] to a uniform nearly plain extension, and in particular to a nearly plain dicompletion. This section contains several results that are important in their own right, and which will also be needed later on.…”

The ditopology generated by pre-open and pre-closed sets, and submaximality in textures

“…On the other hand, in [7,10] compactness and in [11] strong compactness in ditopological texture spaces were introduced including M-prebicontinuous difunctions and preservation of strong compactness and cocompactness however early works in this area started in [3]. In the same direction the notion of real compactness in ditopological texture spaces was introduced in [21] and continued with (real) compactifications, also dicompleteness in [22,23]. The notions of β-open, β-closed sets and β-compactness in ditopological texture spaces were introduced in [9].…”