Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Consonance and topological completeness
in analytic spaces

Author: Ahmed Bouziad
Journal: Proc. Amer. Math. Soc. 127 (1999), 3733-3737
MSC (1991): Primary 54A35; Secondary 54B20, 54C60
Published electronically: May 10, 1999
MathSciNet review: 1610916
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We give a set-valued criterion for a topological space $X$ to be consonant, i.e. the upper Kuratowski topology on the family of all closed subsets of $X$ coincides with the co-compact topology. This characterization of consonance is then used to show that the statement ``every analytic metrizable consonant space is complete'' is independent of the usual axioms of set theory. This answers a question by Nogura and Shakhmatov. It is also proved that continuous open surjections defined on a consonant space are compact covering.

References [Enhancements On Off] (What's this?)

  • [1] Bouziad A., Borel measures in consonant spaces, Top. Appl. 70 (1996), 125-132. MR 97c:54010
  • [2] Dolecki S., G.H. Greco and A. Lechicki, Sur la topologie de la convergence supérieure de Kuratowski, C. R. Acad. Sci. Paris 312 (1991), 923-926. MR 92c:54007
  • [3] Dolecki S., G.H. Greco and A. Lechicki, When do the upper Kuratowski topology (homeomorphically, Scott topology) and the co-compact topology coincide?, Trans. Amer. Math. Soc. 8 (1995), 2869-2884. MR 96c:54010
  • [4] Kanove[??]i V.G. and A.V. Ostrovski[??]i, On non-Borel $F_{II}$-sets, Soviet Math. Dokl. 24 (1981), 386-389.
  • [5] Martin D.A. and R.M. Solovay, Internal Cohen extensions, Ann. Math. Logic 2 (1970), 143-178. MR 42:5787
  • [6] Michael E., A theorem on semi-continuous set-valued functions, Duke Math. Jour. 26 (1959), 647-651. MR 22:229
  • [7] van Mill J., J. Pelant and R. Pol, Selections that characterize topological completeness, Fund. Math., vol. 149, 1996, pp. 127-141. MR 97b:54027
  • [8] Nogura T. and D. Shakhmatov, When does the Fell topology on a hyperspace of closed sets coincide with the meet of the upper Kuratowski and the lower Vietoris topologies?, Top. Appl. 70 (1996), 213-243. MR 97f:54011
  • [9] Pasynkov B.A., On open mappings, Dokl. Akad. Nauk SSSR 175 (1967), 292-295 (in Russian); English transl.: Soviet Math. Dokl. 8 (1967), 853-856. MR 36:862
  • [10] Rudin M.E., Martin's Axiom, in: J. Barwise, ed., Handbook of Mathematical Logic, North-Holland Publishing, 1977, pp. 491-501. MR 56:15351
  • [11] Saint Raymond J., Caractérisation d'espaces Polonais, Sém. Choquet (Initiation Anal.) 5 (1971-1973), 10 p.. MR 57:12811

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 54A35, 54B20, 54C60

Retrieve articles in all journals with MSC (1991): 54A35, 54B20, 54C60

Additional Information

Ahmed Bouziad
Affiliation: Département de Mathématiques, Université de Rouen, CNRS UPRES-A 6085, 76821 Mont Saint-Aignan, France

Keywords: Upper Kuratowski convergence, co-compact topology, analytic spaces, consonant spaces
Received by editor(s): October 7, 1996
Received by editor(s) in revised form: February 10, 1998
Published electronically: May 10, 1999
Communicated by: Alan Dow
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society