Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-29T19:19:04.243Z Has data issue: false hasContentIssue false
  • English
  • Français

P-SPACES AND THE VOLTERRA PROPERTY

Part of: Spaces with richer structures Peculiar spaces

Published online by Cambridge University Press:  31 July 2012

SANTI SPADARO
SANTI SPADARO*
Affiliation:
Department of Mathematics, Faculty of Science and Engineering, York University, Toronto, ON, Canada M3J 1P3 (email: [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.

We study the relationship between generalisations of P-spaces and Volterra (weakly Volterra) spaces, that is, spaces where every two dense Gδ have dense (nonempty) intersection. In particular, we prove that every dense and every open, but not every closed subspace of an almost P-space is Volterra and that there are Tychonoff nonweakly Volterra weak P-spaces. These results should be compared with the fact that every P-space is hereditarily Volterra. As a byproduct we obtain an example of a hereditarily Volterra space and a hereditarily Baire space whose product is not weakly Volterra. We also show an example of a Hausdorff space which contains a nonweakly Volterra subspace and is both a weak P-space and an almost P-space.

Keywords

BaireVolterraP-spacealmost P-spaceweak P-spacedensity topology

MSC classification

Secondary: 54E52: Baire category, Baire spaces 54G10: $P$-spaces
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 87 , Issue 2 , April 2013 , pp. 339 - 345
Copyright
©2012 Australian Mathematical Publishing Association Inc.

References

[1]Baire, R., ‘Sur les fonctions de variables réelles’, Ann. Mat. 3 (1899), 1–123.CrossRefGoogle Scholar
[2]Cao, J. & Junnila, H., ‘When is a Volterra space Baire?’ Topology Appl. 154 (2007), 527–532.CrossRefGoogle Scholar
[3]Dontchev, J., Ganster, M. & Rose, D., ‘Ideal resolvability’, Topology Appl. 93 (1999), 1–16.CrossRefGoogle Scholar
[4]Dunham, W., ‘A historical gem from Vito Volterra’, Math. Mag. 63 (1990), 234–237.CrossRefGoogle Scholar
[5]Gauld, D., Greenwood, S. & Piotrowski, Z., ‘On Volterra spaces III: Topological operations’, Topology Proc. 23 (1998), 167–182.Google Scholar
[6]Gauld, D. & Piotrowski, Z., ‘On Volterra spaces’, Far East J. Math. Sci. 1 (1993), 209–214.Google Scholar
[7]Gruenhage, G. & Lutzer, D., ‘Baire and Volterra spaces’, Proc. Amer. Math. Soc. 128 (2000), 3115–3124.CrossRefGoogle Scholar
[8]Henriksen, M. & Woods, R. G., ‘Weak P-spaces and L-closed spaces’, Questions Answers Gen. Topology 6 (1988), 201–207.Google Scholar
[9]Kunen, K., ‘Weak P-points in ℕ*’, Topology, vol. 23 (Colloq. Math. Soc. János Bolyai, Budapest, 1978), pp. 741–749.Google Scholar
[10]Levy, R., ‘Showering spaces’, Pacific J. Math. 57 (1975), 223–232.CrossRefGoogle Scholar
[11]Levy, R., ‘Almost P-spaces’, Canad. J. Math. 29 (1977), 284–288.CrossRefGoogle Scholar
[12]Luukkainen, J., ‘The density topology is maximally resolvable’, Real Anal. Exchange 25 (1999), 419–420.CrossRefGoogle Scholar
[13]Moors, W., ‘The product of a Baire space with a hereditarily Baire metric space is Baire’, Proc. Amer. Math. Soc. 134 (2006), 2161–2163.CrossRefGoogle Scholar
[14]Rudin, W., ‘Homogeneity problems in the theory of Cech compactifications’, Duke Math. J. 23 (1956), 409–419.CrossRefGoogle Scholar
[15]Tall, F. D., ‘The density topology’, Pacific J. Math. 62 (1976), 275–284.CrossRefGoogle Scholar
[16]Volterra, V., ‘Alcune osservasioni sulle funzioni punteggiate discontinue’, Giornale Mat. 19 (1881), 76–86.Google Scholar
[17]Watson, S., ‘A compact Hausdorff space without P-points in which G δ sets have interior’, Proc. Amer. Math. Soc. 123 (1995), 2575–2577.Google Scholar