Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-22T18:03:30.319Z Has data issue: false hasContentIssue false
  • English
  • Français

Volterra spaces revisited

Part of: Spaces with richer structures Real functions Maps and general types of spaces defined by maps

Published online by Cambridge University Press:  09 April 2009

Jiling Cao  and
David Gauld
Jiling Cao
Affiliation:
Department of MathematicsThe University of AucklandPrivate Bag 92019Auckland 1New Zealand e-mail: [email protected], [email protected]
David Gauld
Affiliation:
Department of MathematicsThe University of AucklandPrivate Bag 92019Auckland 1New Zealand e-mail: [email protected], [email protected]
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 investigate Volterra spaces and relevant topological properties. New characterizations of weakly Volterra spaces are provided. An analogy of the Banach category theorem in terms of Volterra properties is obtained. It is shown that every weakly Volterra homogeneous space is Volterra, and there are metrizable Baire spaces whose hyperspaces of nonempty compact subsets endowed with the Vietoris topology are not weakly Volterra.

Keywords

Baire spacedense Gε-setthe Vietoris topologyVolterra spaceweakly Volterra space.

MSC classification

Secondary: 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) 54C05: Continuous maps 54E52: Baire category, Baire spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 79 , Issue 1 , August 2005 , pp. 61 - 76
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Čoban, M., ‘Note sur la topologie exponentielle’, Fund. Math. 71 (1971), 27–41.CrossRefGoogle Scholar
[2]Fleissner, W. and Kunen, K., ‘Barely Baire spaces’, Fund. Math. 101 (1978), 229–240.CrossRefGoogle Scholar
[3]Gauld, D., ‘Did the young Volterra know about Cantor?’, Math. Magazine 66 (1993), 246–247.CrossRefGoogle Scholar
[4]Gauld, D., Greenwood, S. and Piotrowski, Z., ‘On Volterra spaces II’, Ann. New York Acad. Sci. 806 (1996), 169–173.CrossRefGoogle Scholar
[5]Gauld, D., ‘On Volterra spaces III: Topological operations’, Topology Proc. 23 (1998), 167–182.Google Scholar
[6]Gauld, D. and Piotrowski, Z., ‘On Volterra spaces’, Far East J. Math. Sci. 1 (1993), 209–214.Google Scholar
[7]Gruenhage, G. and Lutzer, D., ‘Baire and Volterra spaces’, Proc. Amer. Math. Soc. 128 (2000), 3115–3124.CrossRefGoogle Scholar
[8]Hankel, H., Untersuchungen über die oscilierenden und unstetigen Funktionen (1870), Ostwalds klassiker der exacten Wissenschaften 153 (Leipzig, 1905).Google Scholar
[9]Haworth, R. C. and McCoy, R. A., Baire spaces, Dissertationes Math. 141 (Instytut Matematyczny, Polska Akademia Nauk, 1977).Google Scholar
[10]Hewitt, E. and Stromberg, K., Real and abstract analysis. A modern treatment of the theory of functions of a real variable (Springer, New York, 1969).Google Scholar
[11]Kunen, K. and Vaughan, J., Handbook of set-theoretic topology (North-Holland, Amsterdam, 1984).Google Scholar
[12]Kuratowski, K., Topology I (Państwowe Wydawnictwo Naukowe, Warsaw, 1966).Google Scholar
[13]McCoy, R. A., ‘Baire spaces and hyperspaces’, Pacific J. Math. 58 (1975), 133–142.CrossRefGoogle Scholar
[14]Mizokami, T., ‘On hyperspaces of generalized metric spaces’, Topology Appl. 76 (1997), 169–173.CrossRefGoogle Scholar
[15]Piotrowski, Z., Roslanowski, A. and Scott, B. M., ‘The pinched-cube topology’, Pacific J. Math. 105 (1983), 319–413.CrossRefGoogle Scholar
[16]Volterra, V., ‘Alcune osservasioni sulle funzioni punteggiate discontinue’, Giornale di Mathematiche 19 (1881), 76–86.Google Scholar