Uniformly discrete hit-and-miss hypertopology. A missing link in hypertopologies

Giuseppe Di Maio, Enrico Meccariello, Somashekhar Naimpally

Abstract

Recently it was shown that the lower Hausdorff metric (uniform) topology is generated by families of uniformly discrete sets as hit sets. This result leads to a new hypertopology which is the join of the above topology and the upper Vietoris topology. This uniformly discrete hit-and-miss hypertopology is coarser than the locally finite hypertopology and finer than both Hausdorff metric (uniform) topology and Vietoris topology. In this paper this new hypertopology is studied. Here is a Hasse diagram in which each arrow goes from a coarser topology to a finer one and equality follows UC or TB as indicated. The diagram clearly shows that the new (underlined) topology provides the missing link.


Keywords

Hypertopology; Hausdorff metric; Vietoris topology; Proximal topology; Proximal locally finite; Hit-and-miss hypertopology; ε-discrete subset; Uniformly discrete; Uniformly isolated; locally finite family

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References

M. Atsuji, Uniform continuity of continuous functions of metric spaces, Pacific J. Math. 8 (1958), 11–16. http://dx.doi.org/10.2140/pjm.1958.8.11

A. Di Concilio, S. Naimpally and P. L. Sharma, Proximal Hypertopologies, Sixth Brazilian Topology Meeting, Campinas, Brazil (1988) [unpublished].

G. Di Maio and L. Holà, On hit-and-miss hyperspace topologies, Rend. Acc. Sci. Fis. Mat. Napoli, (4) 62 (1995), 103–124.

G. Di Maio, E. Meccariello and S. Naimpally, Decomposition of UC spaces, Questions and Answers in Genaral Topology 22 (2004), 13–22.

L. Holà and S. Levi, Decomposition properties of hyperspace topologies, Set-valued Analysis 5 (1997), 309–321. http://dx.doi.org/10.1023/A:1008608209952

V. M. Ivanova, On the theory of spaces of subsets, Dokl. Akad. Nauk. SSSR 101 (1955), 601–603.

J. Keesling, Normality and properties related to compactness in hyperspaces, Proc. Amer. Math. Soc. 24 (1970), 760–766. http://dx.doi.org/10.1090/S0002-9939-1970-0253292-7

M. Marjanovic, Topologies on collections of closed subsets, Publ. Inst. Math. (Beograd) 20 (1966), 125–130.

E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152–182. http://dx.doi.org/10.1090/S0002-9947-1951-0042109-4

J. Nagata, On the uniform topology of bicompactifications, J. Inst. Polytech. Osaka Univ. 1 (1950), 28–38.

S. Naimpally, All hypertopologies are hit-and-miss, Applied General Topology 3 (2002), 45–53.

S. Naimpally and P. L. Sharma, Fine uniformity and the locally finite hyperspace topology, Proc. Amer. Math. Soc. 103 (1988), 641–646. http://dx.doi.org/10.1090/S0002-9939-1988-0943098-9

S. A. Naimpally and B. D.Warrack, Proximity Spaces, Cambridge Tracts inMathematics 59, Cambridge University Press (1970).

F. Wattenberg, Topologies on the set of closed subsets, Pacific J. Math. 68 (1977), 537–551. http://dx.doi.org/10.2140/pjm.1977.68.537

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Cited-By (articles included in Crossref)

This journal is a Crossref Cited-by Linking member. This list shows the references that citing the article automatically, if there are. For more information about the system please visit Crossref site

1. Somashekhar Naimpally, 1931–2014
G. Beer, A. Di Concilio, G. Di Maio, S. Naimpally, C.M. Pareek, J.F. Peters
Topology and its Applications  vol: 188  first page: 97  year: 2015  
doi: 10.1016/j.topol.2015.03.010



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