Strongly Far Proximity and Hyperspace Topology

Year 2020, Volume: 1 Issue: 1, 23 - 29, 31.01.2020

James F. Peters Clara Guadagni

Abstract

This paper introduces strongly far in proximity spaces.
Usually, when we talk about proximities, we mean \textit{Efremovi\v{c} proximities}. Nearness expressions are very useful and also represent a powerful tool because of the relation existing among \textit{Efremovi\v c proximities}, \textit{Weil uniformities} and $\mbox{T}_2$ compactifications. But sometimes \textit{Efremovi\v c proximities} are too strong. So we want to distinguish between a weaker and a stronger forms of proximity. For this reason, we consider at first \textit{Lodato proximity} $\delta$ and then, by this, we define a stronger proximity by using the Efremovi\v{c} property related to proximity.

Keywords

Hit-and-miss topology, Hyperspaces, Proximity, Strongly far

Supporting Institution

Natural Sciences \& Engineering Research Council of Canada (NSERC)

Project Number

185986

Thanks

Many thanks for the invitation to submit this paper.

References

• Di Concilio A., Uniformities, hyperspaces, and normality, Monatsh. Math., 107(3), 303–308, 1989.
• Di Concilio A., Proximity: A powerful tool in extension theory, function spaces, hyperspaces, boolen algebras and point-free geometry, Beyond Topology, F. Mynard, E. Pearl, Eds., Contemporary Mathematics, Amer. Math. Soc., Providence, RI, 486, 89–114, 2009.
• Di Concilio A., Action on hyperspaces, Topology Proc., 41, 85–98, 2013.
• Di Concilio A., Naimpally S.A., Proximal set-open topologies, Boll. Unione Mat. Ital. Sez. B Artic Ric. Mat., 8(1), 173–191, 2000.
• Di Concilio A., Proximal set-open topologies on partial maps, Acta Math. Hungar., 88(3), 227–237, 2000.
• Di Maio G., Naimpally S.A., Comparison of hypertopologies, Rend. Istit. Mat. Univ. Trieste, 22(1–2), 140–161, 1990.
• Efremovic̆V.A., Infinitesimal Spaces (Russian), Dokl. Akad. Nauk SSSR, 76, 341–343, 1951.
• Fell J.M.G., A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space, Proc. Amer. Math. Soc., 13, 472–476, 1962.
• Lodato M.W., On topologically induced generalized proximity relations, Ph.D. Thesis, Rutgers University, 42, 1962.
• Lodato M.W., On topologically induced generalized proximity relations I, Proc. Amer. Math. Soc., 15, 417–422, 1964.
• Lodato M.W., On topologically induced generalized proximity relations II, Pacific J. Math., 17, 131–135, 1966.
• Naimpally S.A., Warrack B.D., Proximity Spaces, Cambridge University Press, Cambridge Tract in Mathematics and Mathematical Physics 59, Cambridge, UK, 1970.
• Naimpally S.A., All hypertopologies are hit-and-miss, App. Gen. Topology, 3, 197–199, 2002.
• Naimpally S.A., Proximity Approach to Problems in Topology and Analysis, Oldenbourg Verlag, Munich, Germany, 2009.
• Peters J.F., Proximal Voronoï regions, convex polygons, & Leader uniform topology, Advances in Math., (4)1, 1–5, 2015.
• Peters J.F., Visibility in proximal Delaunay meshes and strongly near Wallman proximity, Advances in Math., (4)1, 41–47, 2015.
• Peters J.F., Naimpally S.A., Applications of near sets, Notices Amer. Math. Soc., 59(4), 536–542, 2012.
• Peters J.F., Öztürk M.A., Uçkun M., Klee-Phelps convex groupoids, Math. Slovaca, 67(2), 397–400, 2017.
• Vietoris L., Stetige mengen, Monatsch. Math. Phys., 31(1), 173–204, 1921.
• Vietoris L., Bereiche zweiter ordnung, Monatsch. Math. Phys., 32(1), 258–280, 1922.
• Vietoris L., Kontinua zweiter ordnung, Monatsch. Math. Phys., 33(1), 49–62, 1923.
• Vietoris L., Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen, Math. Ann., 97(1), 454–472, 1927.