The ideal generated by σ-nowhere dense sets


  • Jiling Cao University of Auckland
  • Sina Greenwood University of Auckland



Compatible, Ideal, Resolvable, σ-nowhere dense, Volterra, Weakly Volterra


In this paper, we consider the ideal Iσ generated by all σ-nowhere dense sets in a topological space. Properties of this ideal and its relations with the Volterra property are explored. We show that Iσ is compatible with the topology for any given topological space, an analogue to the Banach category theorem. Some applications of this result and the Banach category theorem are also given.


Download data is not yet available.

Author Biographies

Jiling Cao, University of Auckland

Department of Mathematics

Sina Greenwood, University of Auckland

Department of Mathematics


S. Banach, Théorème sur les ensembles de première catégorie, Fund. Math. 16 (1930), 395–398.

J. Cao and D. Gauld, Volterra spaces revisited, J. Austral. Math. Soc. 79 (2005), 61–76.

J. Dontchev, M. Ganster and D. Rose, Ideal resolvability, Topology Appl. 93 (1999), 1–16.

E. K. van Douwen, Applications of maximal topologies, Topology Appl. 51 (1993), 125–139.

D. Gauld, S. Greenwood and Z. Piotrowski, On Volterra spaces III: topological operations, Topology Proc. 23 (1998), 167–182.

I. Guran, Cardinal invariants of paratopological groups, 2nd International Algebraic Conference in Ukraine, Kyiv, Vinnytsia, 1999.

D. Jankovic and T. Hamlett, New topologies from old via ideals, Amer. Math. Monthly 97 (1990), 295–310.

P. Kenderov, I. Kortezov and W. Moors, Topological games and topological groups, Topology Appl. 109 (2001), 157–165.

R. Haworth and R. McCoy, Baire spaces, Dissertationes Math. 141, 1977.

J. Kaniewski and Z. Piotrowski, Concerning continuity apart from a meager set, Proc. Amer. Math. Soc. 98 (1986), 324–328.

J. Kaniewski, Z. Piotrowski and D. Rose, Ideal Banach category theorems, Rocky Mountain J. Math. 28 (1998), 237–251.

K. Kuratowski, Topology, Vol. I, Academic Press, New York, 1966.

O. Ravsky, Paratopological groups II, Math. Stud. 17 (2002), 93–101.

Z. Semadeni, Functions with sets of points of discontinuity belonging to a fixed ideal, Fund. Math. 52 (1963), 26–39.


How to Cite

J. Cao and S. Greenwood, “The ideal generated by σ-nowhere dense sets”, Appl. Gen. Topol., vol. 7, no. 2, pp. 253–264, Oct. 2006.



Regular Articles