Improved Nearness Research

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.

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

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.

Leseberg, D. (2014) Improved Nearness Research.

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.

[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.