Near Compactness of Ditopological Texture Spaces

Abstract

The authors consider the notions of near compactness, near cocompactness, near stability, near costability and near dicompactness in the setting of ditopological texture spaces. In particular preservation of these properties under surjective R-dimaps, co-R-dimaps and bi-R-dimaps is investigated and non-trivial characterizations of near dicompactness are given which generalize those for dicompactness. The notions of semiregularization, semicoregularization and semibiregularization are defined and used to give generalizations of Mrówka's Theorem for near compactness and near cocompactness, and of Tychonoff's Theorem for near compactness, near cocompactness and near dicompactness. Also, results related to pseudo-open and pseudo-closed sets are presented. Finally, examples are given of co-T1 nearly dicompact ditopologies on textures which are not nearly plain and an open question is posed. 2000 AMS Classification: 54 A 05, 54 C 10.

A Tychonoff theorem for graded ditopological texture spaces

“…is an open problem for now as in the ditopological case in [6]. So we use the method which based on the relationship between ditopological and graded ditopological case to prove Theorem 3.16.…”

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

“…The early works on compactness in ditopological texture spaces was begun in [3] and continued in [7,10]. Now let us recall some concepts from [7] which will be needed.…”

“…On the one hand the notion of di-uniformity has been introduced in [17] and continued in [18][19][20], and a textural analogue of the notion of proximity, called a diextremity, was given in [24]. 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 generalization of the Alexander's subbase theorem was given in [7] to prove the Tychonoff theorem for compactness and cocompactness in ditopological texture spaces.…”

Almost-s-Menger ditopological texture spaces

“…The concept of ditopology is more general than topology, bitopology and fuzzy topology. An adequate introduction to the theory of texture spaces and ditopological texture spaces may be obtained from [4][5][6][7][8][9]. For a study of selection principles in ditopological texture spaces see [18,19].…”