Improved Nearness Research

Dieter  Leseberg

doi:10.4236/apm.2014.411070

Dieter Leseberg^*

Abstract: In the realm of Bounded Topology we now consider supernearness spaces as a common generalization of various kinds of topological structures. Among them the so-called Lodato spaces are of significant interest. In one direction they are standing in one-to-one correspondence to some kind of topological extensions. This last statement also holds for contiguity spaces in the sense of Ivanova and Ivanov, respectively and moreover for bunch-determined nearness spaces as Bentley has shown in the past. Further, Do?tch?nov proved that the compactly determined Hausdorff extensions of a given topological space are closely connected with a class of supertopologies which he called b-supertopologies. Now, the new class of supernearness spaces—called paranearness spaces—generalize all of them, and moreover its subclass of clan spaces is in one-to-one correspondence to a certain kind of symmetric strict topological extension. This is leading us to one theorem which generalize all former mentioned.

Keywords: Set-Convergence, Supertopological Space, Lodato Space, Contiguity Space, Nearness, Paranearness

Cite this paper: Leseberg, D. (2014) Improved Nearness Research. Advances in Pure Mathematics, 4, 610-626. doi: 10.4236/apm.2014.411070.

References

[1] Herrlich, H. (1974) A Concept of Nearness. General Topology and Its Applications, 5, 191-212.
http://dx.doi.org/10.1016/0016-660X(74)90021-X

[2] Leseberg, D. (2008) Comprehensive Convergence. IJPAM, 43, 371-392.

[3] Wyler, O. (1988) Convergence of Flters and Ultraflters to Subsets. Lecture Notes in Computer Science, 393, 340-350.

[4] Preuss, G. (2002) Foundations of Topology. An Approach to Convenient Topology. Kluwer, Dordrecht.

[5] Cech, E. (1966) Topological Spaces. Interscience Publishers, London-NY-Sydney.

[6] Tozzi, A. and Wyler, O. (1987) On Categories of Supertopological Spaces. Acta Universitatis Carolinae. Mathematica et Physica, 28, 137-149.

[7] Banaschewski, B. (1964) Extensions of Topological Spaces. Canadian Mathematical Bulletin, 7, 1-23.
http://dx.doi.org/10.4153/CMB-1964-001-5

[8] Bentley, H. (1975) Nearness Spaces and Extension of Topological Spaces. In: Studies in Topology, Academic Press, 47-66.

[9] Choquet, G. (1947) Sur les notions de fltre et de grille. Paris, Ser. A, C.R. Acad. Sci, 224, 171-173.

[10] Doitchinov, D. (1985) Compactly Determined Extensions of Topological Spaces. SERDICA Bulgarice Math. Pub., 11, 269-286.

[11] Deák, J. (1993) Extending a Family of Screens in a Contiguity Space. Topology with Applications, 121-133.

[12] Kent, D. and MIN, W. (2002) Neighbourhood Spaces. IJMMS, 7, 387-399.