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Lightness of Induced Maps and Homeomorphisms

Published online by Cambridge University Press:  20 November 2018

Javier Camargo
Javier Camargo*
Affiliation:
Escuela de Matemáticas, Universidad Industrial de Santander, Ciudad Universitaria, Bucaramanga, Santander, A.A. 678, Colombiae-mail: [email protected]
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Abstract

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An example is given of a map $f$ defined between arcwise connected continua such that $C(f)$ is light and ${{2}^{f}}$ is not light, giving a negative answer to a question of Charatonik and Charatonik. Furthermore, given a positive integer $n$, we study when the lightness of the induced map ${{2}^{f}}$ or ${{C}_{n}}(f)$ implies that $f$ is a homeomorphism. Finally, we show a result in relation with the lightness of $C(C(f))$.

Keywords

54B2054E40light mapsinduced mapscontinuahyperspaces
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 54 , Issue 4 , 01 December 2011 , pp. 607 - 618
Copyright
Copyright © Canadian Mathematical Society 2011

References

[1] Borsuk, K. and Molski, R., On a class of continuous mappings.. Fund. Math. 45(1958), 84–98.Google Scholar
[2] Charatonik, J. J. and Charatonik, W. J., Lightness of induced mappings. Tsukuba J. Math. 22(1998), no. 1, 179–192.Google Scholar
[3] Charatonik, J. J., Illanes, A., and Macías, S., Induced mappings on the hyperspaces Cn (X) of a continuum X. Houston J. Math. 28(2002), no. 4, 781–805.Google Scholar
[4] Hocking, J. G. and Young, G. S., Topology. Second edition. Dover Publications, New York, 1988.Google Scholar
[5] Hosokawa, H., Induced mappings on hyperspaces. Tsukuba J. Math. 21(1997), no. 1, 239–250.Google Scholar
[6] Illanes, A. and Nadler, S. B. Jr., Hyperspaces. Fundamentals and Recent Advances. Monographs and Textbooks in Pure and Applied Mathematics 216, Marcel Dekker, New York, 1999.Google Scholar
[7] Macías, S., Topics on Continua. Pure and Applied Mathematics Series 275, Chapman & Hall/CRC, Boca Raton, FL, 2005.Google Scholar
[8] Maćkowiak, T., Continuos mappings on continua.. Dissertationes Math. (Rozprawy Mat.) 158(1979), 1–95.Google Scholar
[9] Nadler, S. B. Jr., Continuum Theory, An Introduction. Monographs and Textbooks in Pure and Applied Mathematics 158, Marcel Dekker, New York, 1992.Google Scholar
[10] Nadler, S. B. Jr., Dimension Theory: An Introduction with Exercises. Aportaciones Matemáticas. Textos 18. Sociedad Matemática Mexicana. México, 2002.Google Scholar
[11] Nadler, S. B. Jr., Hyperspaces of Sets. A Text with Research Questions. Aportaciones Matemáticas, Textos 33, Sociedad Matemática Mexicana, 2006.Google Scholar