On [ϱ1, ϱ2]-Lower Superdense Sets

The main goal of the paper is to characterize the families of [ϱ₁, ϱ₂]-lower superdense subsets of ℝ, which generalize well-known notions of density topology (in our paper, denoted by L₁^♦)and T^∗ topology. Dense sets preserve density, i.e., if E ∈ L₁^♦, then for every measurable F ⊂ ℝ, E ∩ F possesses the same lower and upper density as F at every x ∈ E ∩ F. On the other hand, elements of T^∗ preserve positive lower density, i.e., if E ∈ T^∗, F is measurable, x ∈ E∩F and d (F, x) > 0, then d (E ∩ F, x) > 0. Taking arbitrary 0 ≤ ϱ₁ ≤ ϱ₂ ≤ 1, ϱ₂ − ϱ₁ < 1, we can define subsets E ⊂ ℝ which preserve [ϱ₁, ϱ₂]-lower density, i.e., if F is measurable, x ∈ E ∩ F and d (F, x) is greater or not less than ϱ₂,then d (E ∩ F, x) is greater or not less than ϱ₁. We can define four types of superdense sets, but three of them are equal. Even though the definition and properties of [ϱ₁, ϱ₂]-lower superdensity and T^∗ topology are similar and all of them consist of very big sets, these families are essentially different. In the paper, we focus on basic properties, characterizations of superdense sets and relationships between [ϱ₁, ϱ₂]-lower superdense sets for different indices [ϱ₁, ϱ₂]. We apply the notion of [ϱ₁, ϱ₂]-lower superdensity to find adders of ϱ-lower continuous functions.