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Openness of induced projections

Author(s): Janusz J. Charatonik; Wlodzimierz J. Charatonik; Alejandro Illanes
Journal: Proc. Amer. Math. Soc. 128 (2000), 3711-3714.
MSC (2000): Primary 54B20, 54E40, 54F15, 54F50, 54F65
Posted: June 7, 2000
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Abstract: For continua $X$ and $Y$ it is shown that if the projection $f:X\times Y\to X$ has its induced mapping $C(f)$ open, then $X$ is $C^{*}$-smooth. As a corollary, a characterization of dendrites in these terms is obtained.

References:

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J. J. Charatonik, W. J. Charatonik and A. Illanes, Openness of induced mappings, Topology Appl. 98 (1999), 67-80. CMP 99:17
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J. Grispolakis, S. B. Nadler, Jr., and E. D. Tymchatyn, Some properties of hyperspaces with applications to continua theory, Canad. J. Math. 31 (1979), 197-210. MR 80j:54029
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H. Hosokawa, Induced mappings between hyperspaces II, Bull. Tokyo Gakugei Univ. (4) 44 (1992), 1-7. MR 94a:54036
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H. Hosokawa, Induced mappings on hyperspaces, Tsukuba J. Math. 21 (1997), 239-250. MR 98f:54011
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K. Kuratowski, Topology, vol. 2, Academic Press and PWN, New York, London and Warszawa, 1968. MR 41:4467
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S. B. Nadler, Jr., Hyperspaces of sets, M. Dekker, 1978. MR 58:18330

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Additional Information:

Janusz J. Charatonik
Affiliation: Mathematical Institute, University of Wroclaw, pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland - Instituto de Matemáticas, UNAM, Circuito Exterior, Ciudad Universitaria, 04510 México, D. F., México
Email: jjc@hera.math.uni.wroc.pl, jjc@gauss.matem.unam.mx

Wlodzimierz J. Charatonik
Affiliation: Mathematical Institute, University of Wroclaw, pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland - Departamento de Matemáticas, Facultad de Ciencias, UNAM, Circuito Exterior, Ciudad Universitaria, 04510 México, D. F., México
Address at time of publication: Department of Mathematics and Statistics, University of Missouri-Rolla, Rolla, Missouri 65409-0020
Email: wjcharat@hera.math.uni.wroc.pl, wjcharat@lya.fciencias.unam.mx

Alejandro Illanes
Affiliation: Instituto de Matemátic as, UNAM, Circuito Exterior, Ciudad Universitaria, 04510 México, D. F., México
Email: illanes@gauss.matem.unam.mx

DOI: 10.1090/S0002-9939-00-05589-1
PII: S 0002-9939(00)05589-1
Keywords: $C^{*}$-smooth, continuum, dendrite, induced mapping, open, projection
Received by editor(s): February 2, 1999
Posted: June 7, 2000
Communicated by: Alan Dow
Copyright of article: Copyright 2000, American Mathematical Society

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