Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-05-13T12:02:20.524Z Has data issue: false hasContentIssue false
  • English
  • Français

Reflexive open mappings on generalized graphs

Part of: Special properties Maps and general types of spaces defined by maps

Published online by Cambridge University Press:  09 April 2009

Stanislaw Miklos
Stanislaw Miklos
Affiliation:
Institute of Mathematics University of Wroclawpl. Grunwaldzki 2/4 50–384 Wroclaw, Poland
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we show that a locally connected and locally compact metric image of a generalized graph under a reflexive open mapping is a generalized graph; further, we characterize all acyclic generalized graphs X with the property that any locally one-to-one reflexive open mapping of X into a Hausdorff space is globally one-to-one. Several problems are posed and some examples are given.

Keywords

Hausdorff spaceslocally connected and locally compact metric spacesgeneralized graphreflexive open mappingsone-to-one mappings

MSC classification

Secondary: 54C10: Special maps on topological spaces (open, closed, perfect, etc.) 54F50: Spaces of dimension $leq 1$; curves, dendrites 54F65: Topological characterizations of particular spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 47 , Issue 3 , December 1989 , pp. 343 - 349
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Charatonik, J. J. and Miklos, S., ‘Generalized graphs and their open mappings’, Rend. Mat. 2 (1982), 335–354.Google Scholar
[2]Charatonik, J. J. and Miklos, S., ‘A characterization of cyclic graphs,’ Bull. Acad. Polon. Sci. Sér. Sci. Math. 30 (1982), 453–455.Google Scholar
[3]Duda, E., ‘One to one mappings and applications’, General topology and its applications 1, pp. 135–142 (North-Holland Publishing Company, 1971).Google Scholar
[4]Duda, E. and Smith, J. W., ‘Reflexive open mappings’, Pacific J. Math. 38 (1971), 597–611.CrossRefGoogle Scholar
[5]Eberhart, C. A., Fugate, J. B., and Gordh, G. R. Jr, ‘Branchpoint covering theorems for confluent and weakly confluent maps’, Proc. Amer. Math. Soc. 55 (1976), 409–415.CrossRefGoogle Scholar
[6]Jones, F. B., ‘One-to-one continuous images of a line’, General topology and its relations to modern analysis and algebra, Proceedings of the Kanpur Topological Conference, 1968, pp. 157–160.Google Scholar
[7]Jungck, G. F., ‘Local homeomorphisms’, Dissertationes Math. (Rozprawy Mat.) 209 (1983).Google Scholar
[8]Kuratowski, K., Topology 2 (PWN, Warszawa, 1968).Google Scholar
[9]Lelek, A. and McAuley, L. F., ‘On hereditarily locally connected spaces and one-to-one continuous images of a line’, Colloq. Math. 17 (1967), 319–324.CrossRefGoogle Scholar
[10]Miklos, S., ‘Local homeomorphisms onto cyclic graphs’, Bull. Acad. Polon. Sci. Sér. Sci. Math. 31 (1983), 381–384.Google Scholar
[11]Whyburn, G. T., Analytic topology (Amer. Math. Soc. Colloq. Publ., vol. 28, Providence, R.I., 1942).Google Scholar
[12]Whyburn, G. T., ‘Open and closed mappings’, Duke Math. J. 17 (1950), 69–74.CrossRefGoogle Scholar