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

• Di Maio, Giuseppe ^[1] ; Meccariello, Enrico ^[2] ; Naimpally, Somashekhar
  1. [1] Seconda Università degli Studi di Napoli
  2. [2] Università del Sannio
• Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 7, Nº. 2, 2006, págs. 245-252
• Idioma: inglés
• DOI: 10.4995/agt.2006.1927
• Enlaces
  • Texto completo
• Resumen

  • 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.

• Referencias bibliográficas
  • 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S. Naimpally and P. L. Sharma, Proximal Hypertopologies, Sixth Brazilian Topology Meeting, Campinas, Brazil (1988) [unpublished].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