On abstract and almost-abstract density topologies

Generalized Densities on ℝ n and their Applications

“…Since any (ψ, n)-density point is an ordinary density point, then for any set A ∈ L n the difference Φ (ψ,n) (A)\A is a set of measure zero. Hence, the operator Φ (ψ,n) is an almost lower density operator (see [14]). It is not a lower density operator.…”

“…From the fact that the operator Φ (ψ,n) is an almost lower density operator, we immediately obtain (for details, see [14]) that (R n , T (ψ,n) ) is neither a first countable, nor a second countable, nor a separable, nor a Lindelöf space. Several other properties of T (ψ,n) follow from the proofs carried for the topology T ψ .…”

On lower density type operators and topologies generated by them

“…It turns out that in none of these families there is the smallest and the largest topology with respect to the relation ⊂. In the case of the smallest topology, the appropriate justification could be found in [6]. Now, we would like to concentrate on the smallest topology containing the union of all abstract density topologies on R, L, L .…”

“…In [6] one can find that in a measurable space X, S, J such that J = X, the existence of the smallest topology in the family T ordered by relation ⊂ is equivalent to the equality S = S 0 . The existence of the largest topology in the family T ordered by the relation ⊂ is also connected with the analogous condition.…”

“…We will investigate, among others, the existence of the largest topology with respect to the inclusion. The existence of the smallest with respect to the relation ⊂ topology generated by lower and almost-lower density operators was investigated in [6]. Moreover, some other relations in the family of all topologies generated by operators mentioned above were studied in [10].…”