On Density Topologies with Respect to Invariant σ-Ideals

Abstract

Abstract. The density topologies with respect to measure and category are motivation to consider the density topologies with respect to invariant σ-ideals on R. The properties of such topologies, including the separation axioms, are studied. NotationBy R we shall denote the set of all reals numbers and by N the set of positive integers. Let l stand for Lebesgue measure. The capitals L and L denote the σ-algebra of all Lebesgue measurable sets in R and the σ-ideal of all Lebesgue null sets. The natural topology on R is denoted by T 0 . If T is a topology on R, then we fix the notation: B(T ) -the σ-algebra of all Borel sets with respect to T , Ba(T ) -the σ-algebra of all sets having the Baire property with respect to T , K(T ) -the σ-ideal of all meager sets with respect to T .2000 Mathematics Subject Classification. 28A05, 54A10. Key words and phrases. Density point, density topology, the separation axioms, invariant ideals and algebras.

Pointwise density topology

“…In the proof of Theorem 3.6, we need the following lemma, of which the proof is similar to that of Lemma 2.7 in [3], but for the convenience of the reader we include it here. Proof.…”

On the regularity of topologies in the family of sets having the Baire property

“…If T Φ is a topology then we say that T Φ is generated by Φ. Some ideas of this approach one can find in [4]. It is well observing the following fact.…”

On the abstract density topologies generated by lower and almost lower density operators

“…• density topology with respect to the O'Malley points (W. Poreda, W. Wilczyński (2001), see [23]); • density topology with respect to measure and category (J. Hejduk (2002), see [10]); • complete density topology (W. Wilczyński, W. Wojdowski (2007), see [32]); • f -density topology (M. Filipczak, T. Filipczak, (2008), see [3]);…”