Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

Cut-point spaces

Author(s): B. Honari; Y. Bahrampour
Journal: Proc. Amer. Math. Soc. 127 (1999), 2797-2803.
MSC (1991): Primary 54F15, 54F65
Posted: April 15, 1999
MathSciNet review: 1600152
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Abstract: The notion of a cut-point space is introduced as a connected topological space without any non-cut point. It is shown that a cut-point space is infinite. The non-cut point existence theorem is proved for general (not necessarily ) topological spaces to show that a cut-point space is non-compact. Also, the class of irreducible cut-point spaces is studied and it is shown that this class (up to homeomorphism) has exactly one member: the Khalimsky line.

References:

[1]: J. G. Hocking, G.S. Young, Topology, Addison-Wesley (1961). MR 23:A2857
[2]: E.D. Khalimsky, R. Kopperman, P.R. Meyer, ``Computer graphics and connected topologies on finite ordered sets'', Topology and its Applications, 36 (1990) 1-17. MR 92c:54037
[3]: R. Kopperman, ``The Khalimsky line in digital topology'', in Y.-L. O et al., eds., Shape in Picture: Mathematical Description of Shape in Grey-Level Images, Springer-Verlag (1994) 3-20.
[4]: R. L. Moore, ``Concerning simple continuous curves'', Trans. Amer. Math. Soc., 21 (1920) 333-347.
[5]: S. B. Nadler, Jr., Continuum Theory: An Introduction, Marcel Dekker (1992). MR 93m:54002
[6]: G.T. Whyburn, ``Cut points in general topological spaces'', Proc. Nat. Acad. Sci., USA, 61 (1968) 380-387. MR 39:3463

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Additional Information:

B. Honari
Affiliation: Faculty of Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran
Email: honari@arg3.uk.ac.ir

Y. Bahrampour
Affiliation: Faculty of Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran
Email: bahram@arg3.uk.ac.ir

DOI: 10.1090/S0002-9939-99-04839-X
PII: S 0002-9939(99)04839-X
Keywords: Continuum, cut-point space, cut point, non-cut point existence theorem, irreducible cut-point space, Khalimsky line
Received by editor(s): March 3, 1997
Received by editor(s) in revised form: November 20, 1997
Posted: April 15, 1999
Communicated by: Alan Dow
Copyright of article: Copyright 1999, American Mathematical Society

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