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

Abstract

The paper concerns the topologies introduced in the family of sets having the Baire property in a topological space (X, τ) and in the family generated by the sets having the Baire property and given a proper σ-ideal containing τ-meager sets. The regularity property of such topologies is investigated.

On equivalence of topological and restrictional continuity

“…By Corollary 3 in [1] we obtain the following theorem giving equivalence of topological and restrictional continuity on residual sets.…”

“…The following theorems are a special case of similar theorems in [1] concerning arbitrary topological Baire spaces.…”

On lower density type operators and topologies generated by them

“…On the other hand the family T Φ is not a topology. To see that it is sufficient to consider a set D ⊂ B and D L. It is worth adding that in general there exists an operator Φ ∈ SLDO( X, S, J ) not generating a topology if and only if S 0 S 2 X (see [4]).…”

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

“…Theorem 25.37 (cf. [11]). Let X, τ be a topological space such that X / ∈ M(τ) and Φ be an almost lower density operator on (X, B(τ), M(τ)).…”