Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Relations approximated by continuous functions


Authors: L'. Holá and R. A. McCoy
Journal: Proc. Amer. Math. Soc. 133 (2005), 2173-2182
MSC (2000): Primary 54C35, 54B20, 54C08
DOI: https://doi.org/10.1090/S0002-9939-05-07793-2
Published electronically: February 15, 2005
MathSciNet review: 2137885
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $X$ be a Tychonoff space, let $C(X)$ be the space of all continuous real-valued functions defined on $X$ and let $CL(X \times R)$ be the hyperspace of all nonempty closed subsets of $X\times R$. We prove the following result. Let $X$ be a locally connected, countably paracompact, normal $q$-space without isolated points, and let $F \in CL(X \times R)$. Then $F$ is in the closure of $C(X)$ in $CL(X \times R)$ with the locally finite topology if and only if $F$is the graph of a cusco map. Some results concerning an approximation in the Vietoris topology are also given.


References [Enhancements On Off] (What's this?)

  • [Be1] G. Beer, Topologies on closed and closed convex sets, Kluwer Academic Publisher, 1993. MR 1269778 (95k:49001)
  • [Be2] G. Beer, The approximation of real functions in the Hausdorff metric, Houston J. of Math. 10 (1984), 325-338. MR 0763235 (85m:41045)
  • [Be3] G. Beer, On functions that approximate relations, Proc. Amer. Math. Soc. 88 (1983), 643-647. MR 0702292 (84m:54018)
  • [Be4] G. Beer, On a theorem of Cellina for set valued functions, Rocky Mountain J. of Math. 18 (1988), 37-47. MR 0935726 (89c:54034)
  • [BHPV] G. Beer, J. Himmelberg, K. Prikry and F.S. Van Vleck, The locally finite topology on $2^{X}$, Proc. Amer. Math. Soc. 101 (1987), 168-172. MR 0897090 (88f:54014)
  • [Bo] J.M. Borwein, Minimal CUSCOS and Subgradients of Lipschitz Functions, Fixed point theory and applications (Marseille, 1989), 57-81, Pitman Res. Notes Math. Ser., 252, Longman Sci. Tech., Harlow, 1991. MR 1122818 (92j:46077)
  • [Ce] A. Cellina, A further result on the approximation of set valued mappings, Rendiconti Acc. Naz. Lincei 48 (1970), 412-416. MR 0276935 (43:2675)
  • [DB] F. De Blasi, Characterizations of certain classes of semicontinuous multifunctions by continuous approximation, J. Math. Anal. Appl. 106 (1985), 1-18. MR 0780314 (86h:54021)
  • [DBM] F. De Blasi and J. Myjak, On continuous approximations for multifunctions, Pacific J. Math. 123 (1986), 9-31. MR 0834135 (87g:54047)
  • [DHP] G. Di Maio, L'. Holá, J. Pelant, Properties related to the first countability of hyperspace topologies, Questions and Answers in General Topology 19 (2001), 139-157. MR 1815355 (2002a:54004)
  • [Ch] J.P.R. Christensen, Theorems of Namioka and R.E. Johnson type for upper semicontinuous and compact valued set-valued mappings, Proc. Amer. Math. Soc. 86 (1982), 649-655. MR 0674099 (83k:54014)
  • [En] R. Engelking, General Topology, Helderman, Berlin, 1989.
  • [Ho1] L'. Holá, On relations approximated by continuous functions, Acta Universitatis Carolinae - Mathematica et Physica 28 (1987), 67-72. MR 0932741 (89e:54031)
  • [Ho2] L'. Holá, Hausdorff metric on the space of upper semicontinuous multifunctions, Rocky Mountain J. Math. 22 (1992), 601-610. MR 1180723 (93j:54010)
  • [Hu] M. Hukuhara, Sur l' application semi-continue dont la valeur est un compact convexe, Funkcial. Ekvac. 10 (1967), 43-66. MR 0222856 (36:5906)
  • [Mc] R.A. McCoy, Densely continuous forms in Vietoris hyperspaces, Set-Valued Analysis 8 (2000), 267-271. MR 1790485 (2001h:54017)
  • [MN] R.A. McCoy and I. Ntantu, Topological Properties of Spaces of Continuous Functions, Springer-Verlag, Berlin, 1988. MR 0953314 (90a:54046)
  • [Mi] E. Michael, A note on closed maps and compact sets, Isreal J. Math. 2 (1964), 173-176. MR 0177396 (31:1659)
  • [NS] S.A. Naimpally and P.L. Sharma, Fine uniformity and the locally finite hyperspace topology, Proc. Amer. Math. Soc. 103 (1988), 641-646. MR 0943098 (89f:54021)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 54C35, 54B20, 54C08

Retrieve articles in all journals with MSC (2000): 54C35, 54B20, 54C08


Additional Information

L'. Holá
Affiliation: Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava, Slovakia
Email: hola@mat.savba.sk

R. A. McCoy
Affiliation: Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061
Email: mccoy@math.vt.edu

DOI: https://doi.org/10.1090/S0002-9939-05-07793-2
Keywords: Set-valued mapping, Vietoris topology, locally finite topology, upper-semicontinuous multifunction, usco map, cusco map
Received by editor(s): October 14, 2003
Received by editor(s) in revised form: April 8, 2004
Published electronically: February 15, 2005
Communicated by: Alan Dow
Article copyright: © Copyright 2005 American Mathematical Society

American Mathematical Society