## Dendrites and light mappings

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- by Janusz J. Charatonik and Paweł Krupski PDF
- Proc. Amer. Math. Soc.
**132**(2004), 1211-1217 Request permission

## Abstract:

It is shown that a metric continuum $X$ is a dendrite if and only if for every compact space (continuum) $Y$ and for every light confluent mapping $f: Y \to f(Y)$ such that $X \subset f(Y)$ there is a copy $X’$ of $X$ in $Y$ for which the restriction $f|X’: X’ \to X$ 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 $f$ are also discussed. This is a continuation of a previous study by the authors (2000), where open mappings $f$ were considered.## References

- Janusz J. Charatonik, Włodzimierz J. Charatonik, and PawełKrupski,
*Dendrites and light open mappings*, Proc. Amer. Math. Soc.**128**(2000), no. 6, 1839–1843. MR**1751997**, DOI 10.1090/S0002-9939-00-05693-8 - J. J. Charatonik, W. J. Charatonik, and S. Miklos,
*Confluent mappings of fans*, Dissertationes Math. (Rozprawy Mat.)**301**(1990), 86. MR**1055706** - J. J. Charatonik and K. Omiljanowski,
*On light open mappings*, Baku International Topological Conference Proceedings, ELM, Baku, 1989, pp. 211–219. - R. Engelking and A. Lelek,
*Metrizability and weight of inverses under confluent mappings*, Colloq. Math.**21**(1970), 239–246. MR**263041**, DOI 10.4064/cm-21-2-239-246 - W. T. Ingram,
*$C$-sets and mappings of continua*, Proceedings of the 1982 Topology Conference (Annapolis, Md., 1982), 1982, pp. 83–90. MR**696623** - Józef Krasinkiewicz,
*Path-lifting property for 0-dimensional confluent mappings*, Bull. Polish Acad. Sci. Math.**48**(2000), no. 4, 357–367. MR**1797901** - K. Kuratowski,
*Topology. Vol. II*, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1968. New edition, revised and augmented; Translated from the French by A. Kirkor. MR**0259835** - A. Lelek and David R. Read,
*Compositions of confluent mappings and some other classes of functions*, Colloq. Math.**29**(1974), 101–112. MR**367900**, DOI 10.4064/cm-29-1-101-112 - T. Maćkowiak,
*Continuous mappings on continua*, Dissertationes Math. (Rozprawy Mat.)**158**(1979), 95. MR**522934** - Tadeusz Maćkowiak,
*Singular arc-like continua*, Dissertationes Math. (Rozprawy Mat.)**257**(1986), 40. MR**881290** - T. Maćkowiak and E. D. Tymchatyn,
*Some properties of open and related mappings*, Colloq. Math.**49**(1985), no. 2, 175–194. MR**830801**, DOI 10.4064/cm-49-2-175-194 - T. Maćkowiak and E. D. Tymchatyn,
*Some classes of locally connected continua*, Colloq. Math.**52**(1987), no. 1, 39–52. MR**891496**, DOI 10.4064/cm-52-1-39-52 - J. Mioduszewski,
*On continuous selections of multi-valued functions on dendrites*, Prace Mat.**5**(1961), 73–77 (Polish, with Russian and English summaries). MR**0130674** - S. B. Nadler, Jr.,
*Continua determined by surjections of various types*, preprint. - J. J. Corliss,
*Upper limits to the real roots of a real algebraic equation*, Amer. Math. Monthly**46**(1939), 334–338. MR**4**, DOI 10.1080/00029890.1939.11998880

## Additional Information

**Janusz J. Charatonik**- Affiliation: Instituto de Matemáticas, UNAM, Circuito Exterior, Ciudad Universitaria, 04510 México, D. F., México
- Email: jjc@matem.unam.mx
**Paweł Krupski**- Affiliation: Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
- Email: krupski@math.uni.wroc.pl
- 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
- © Copyright 2003 American Mathematical Society
- 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
- MathSciNet review: 2045440