Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-28T14:43:04.019Z Has data issue: false hasContentIssue false

Selections of multivalued maps and shape domination

Published online by Cambridge University Press:  24 October 2008

José M. R. Sanjurjo
Affiliation:
Departamento de Geometria y Topologia, Facultad de Matematicas, Universidad Complutense, 28040 Madrid, Spain

Abstract

Some results are presented which establish connections between shape theory and the theory of multivalued maps. It is shown how to associate an upper-semi-continuous multivalued map F: XY to every approximative map f = {fk, XY} in the sense of K. Borsuk and it is proved that, in certain circumstances, if F is ‘small’ and admits a selection, then the shape morphism S(f) is generated by a map, and if F admits a coselection then S(f) is a shape domination.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1990

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Addis, D. F. and Gresham, J. H.. A class of infinite dimensional spaces. Part 1: Dimension theory and Alexandroff's problem. Fund. Math. 101 (1978), 195205.CrossRefGoogle Scholar
[2]Borsuk, K.. Concerning homotopy properties of compacta. Fund. Math. 62 (1968), 223254.CrossRefGoogle Scholar
[3]Borsuk, K.. Theory of Shape. Monogr. Mat. no. 59 (Polish Scientific Publishers, 1975).Google Scholar
[4]Borsuk, K.. Some quantitative properties of shapes. Fund. Math. 93 (1976), 197212.CrossRefGoogle Scholar
[5]Ĉerin, Z.. Homotopy properties of locally compact spaces at infinity-calmness and smoothness. Pacific J. Math. 79 (1978), 6991.CrossRefGoogle Scholar
[6]Ĉerin, Z. and Sostak, A. P.. Some remarks on Borsuk's fundamental metric. In Proceedings Colloquium on Topology, Budapest 1978, Colloq. Soc. Janos Bolyay no. 23 (North-Holland, 1980). pp. 233252.Google Scholar
[7]Ĉerin, Z. and Watanabe, T.. Borsuk fixed point theorem for multivalued maps. In Geometric Topology and Shape Theory (eds. Mardešić, S. and Segal, J.), Lecture Notes in Math. vol. 1283 (Springer-Verlag, 1987), pp. 3037.CrossRefGoogle Scholar
[8]Dydak, J. and Segal, J.. Shape Theory: An Introduction. Lecture Notes in Math. vol. 688 (Springer-Verlag, 1978).CrossRefGoogle Scholar
[9]Haver, W. E.. A covering property for metric spaces. In Proceedings of Topology Conference (eds. Dickman, R. F. and Hatcher, P.), Lectures Notes in Math. vol. 375 (Springer-Verlag 1974), pp. 108113.CrossRefGoogle Scholar
[10]Kodama, Y.. Multivalued maps and shape. Glasnik Mat. 12 (32) (1977), 133142.Google Scholar
[11]Koyama, A.. Various compact multi-retracts and shape theory. Tsukuba J. Math. 6 (1982), 319332.CrossRefGoogle Scholar
[12]Lisica, J. T.. Strong shape theory and multivalued maps. Glasnik Mat. 18 (38) (1983), 371382.Google Scholar
[13]Mardešić, S. and Segal, J.. Shape Theory (North Holland, 1982).Google Scholar
[14]Sanjurjo, J. M. R.. On quasi-domination of compacta. Colloq. Math. 48 (1984), 213217.CrossRefGoogle Scholar
[15]Spiez, S.. Movability and uniform movability. Bull. Acad. Polon. Sci. Math. 22 (1974), 4345.Google Scholar
[16]Suszycki, A.. Retracts and homotopies for multi-maps. Fund. Math. 95 (1983), 926.CrossRefGoogle Scholar