Dendrites and light mappings
Authors:
Janusz J. Charatonik and Pawel Krupski
Journal:
Proc. Amer. Math. Soc. 132 (2004), 1211-1217
MSC (2000):
Primary 54C60, 54C65, 54E40, 54F50
DOI:
https://doi.org/10.1090/S0002-9939-03-07270-8
Published electronically:
October 29, 2003
MathSciNet review:
2045440
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: It is shown that a metric continuum is a dendrite if and only if for every compact space (continuum)
and for every light confluent mapping
such that
there is a copy
of
in
for which the restriction
is a homeomorphism. As a corollary it follows that only dendrites have the lifting property with respect to light confluent mappings. Other classes of mappings
are also discussed. This is a continuation of a previous study by the authors (2000), where open mappings
were considered.
- 1. J. J. Charatonik, W. J. Charatonik, and P. Krupski, Dendrites and light open mappings, Proc. Amer. Math. Soc. 128 (2000), 1839-1843. MR 2001c:54027
- 2. J. J. Charatonik, W. J. Charatonik, and S. Miklos, Confluent mappings of fans, Dissertationes Math. (Rozprawy Mat.) 301 (1990), 86 pp. MR 91h:54056
- 3. J. J. Charatonik and K. Omiljanowski, On light open mappings, Baku International Topological Conference Proceedings, ELM, Baku, 1989, pp. 211-219.
- 4. R. Engelking and A. Lelek, Metrizability and weight of inverses under confluent mappings, Colloq. Math. 21 (1970), 239-246. MR 41:7646
- 5.
W. T. Ingram,
-sets and mappings of continua, Topology Proc. 7 (1982), 83-90. MR 85i:54040
- 6.
J. Krasinkiewicz, Path-lifting property for
-dimensional confluent mappings, Bull. Polish Acad. Sci. Math. 48 (2000), 357-367. MR 2001h:54020
- 7. K. Kuratowski, Topology, vol. 2, Academic Press, New York, London and PWN Polish Scientific Publishers, Warsaw, 1968. MR 41:4467
- 8. A. Lelek and D. R. Read, Compositions of confluent mappings and some other classes of functions, Colloq. Math. 29 (1974), 101-112. MR 51:4142
- 9. T. Mackowiak, Continuous mappings on continua, Dissertationes Math. (Rozprawy Mat.) 158 (1979), 95 pp. MR 81a:54034
- 10. T. Mackowiak, Singular arc-like continua, Dissertationes Math. (Rozprawy Mat.) 257 (1986), 40 pp. MR 88f:54066
- 11. T. Mackowiak and E. D. Tymchatyn, Some properties of open and related mappings, Colloq. Math. 49 (1985), 175-194. MR 87g:54038
- 12. T. Mackowiak and E. D. Tymchatyn, Some classes of locally connected continua, Colloq. Math. 52 (1987), 39-52. MR 88h:54047
- 13. J. Mioduszewski, Twierdzenie o selektorach funkcyj wielowartosciowych na dendrytach [A theorem on the selectors of multi-valued functions on dendrites], Prace Mat. 5 (1961), 73-77, in Polish; Russian and English summaries. MR 24:A534
- 14. S. B. Nadler, Jr., Continua determined by surjections of various types, preprint.
- 15. G. T. Whyburn, Analytic topology, American Mathematical Society Colloquium Publications, Vol. 28, Providence, RI, 1942, reprinted with corrections 1971. MR 4:86b
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Additional Information
Janusz J. Charatonik
Affiliation:
Instituto de Matemáticas, UNAM, Circuito Exterior, Ciudad Universitaria, 04510 México, D. F., México
Email:
jjc@matem.unam.mx
Pawel Krupski
Affiliation:
Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
Email:
krupski@math.uni.wroc.pl
DOI:
https://doi.org/10.1090/S0002-9939-03-07270-8
Keywords:
Confluent,
continuum,
dendrite,
lifting,
light,
mapping,
open
Received by editor(s):
March 14, 2001
Received by editor(s) in revised form:
February 4, 2002
Published electronically:
October 29, 2003
Communicated by:
Alan Dow
Article copyright:
© Copyright 2003
American Mathematical Society