Journal of the Mathematical Society of Japan

Approaching points by continuous selections

Valentin GUTEV

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Some further results about special Vietoris continuous selections and totally disconnected spaces are obtained, also several applications are demonstrated. In particular, it is demonstrated that a homogeneous separable metrizable space has a continuous selection for its Vietoris hyperspace if and only if it is discrete,or a discrete sum of copies of the Cantor set, or is the irrational numbers.

Article information

J. Math. Soc. Japan, Volume 58, Number 4 (2006), 1203-1210.

First available in Project Euclid: 21 May 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 54B20: Hyperspaces 54C65: Selections [See also 28B20]
Secondary: 54F65: Topological characterizations of particular spaces

hyperspace topology Vietoris topology continuous selection


GUTEV, Valentin. Approaching points by continuous selections. J. Math. Soc. Japan 58 (2006), no. 4, 1203--1210. doi:10.2969/jmsj/1179759545.

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  • P. Alexandroff and P. Urysohn, Über nulldimensionale Punktmengen, Math. Ann., 98 (1928), 89–106.
  • G. Artico, U. Marconi, J. Pelant, L. Rotter and M. Tkachenko, Selections and suborderability, Fund. Math., 175 (2002), 1–33.
  • D. Bertacchi and C. Costantini, Existence of selections and disconnectedness properties for the hyperspace of an ultrametric space, Topology Appl., 88 (1998), 179–197.
  • L. E. J. Brouwer, On the structure of perfect sets of points, Proc. Akad. Amsterdam, 12 (1910), 785–794.
  • C. Costantini and V. Gutev, Recognizing special metrics by topological properties of the “metric”-proximal hyperspace, Tsukuba J. Math., 26 (2002), 145–169.
  • V. Gutev and T. Nogura, Selections for Vietoris-like hyperspace topologies, Proc. London Math. Soc., 80 (2000), 235–256.
  • V. Gutev and T. Nogura, Vietoris continuous selections and disconnectedness-like properties, Proc. Amer. Math. Soc., 129 (2001), 2809–2815.
  • V. Gutev and T. Nogura, Some problems on selections for hyperspace topologies, Applied General Topology, 5 (2004), 71–78.
  • V. Gutev and T. Nogura, Selection pointwise-maximal spaces, Topology Appl., 146–147 (2005), 397–408.
  • Y. Hattori and T. Nogura, Continuous selections on certain spaces, Houston J. Math., 21 (1995), 585–594.
  • J. van Mill, J. Pelant and R. Pol, Selections that characterize topological completeness, Fund. Math., 149 (1996), 127–141.
  • D. Montgomery and L. Zippin, Topological transformation groups, Interscience, New York, 1955.
  • P. J. Nyikos and H.-C. Reichel, Topologically orderable spaces, Gen. Top. Appl., 5 (1975), 195–204.
  • M. Venkataraman, M. Rajagopalan and T. Soundararajan, Orderable topological spaces, Gen. Top. Appl., 2 (1972), 1–10.