Interior in the simple density topology

摘要

We consider a simple density topology T_s on the real line generated by the operator Φ_s. Let us define the operator Φ_s~α for a countable ordinal α in the following way: Φ_s~α (E) = Φ(Φ_s~β (E)) if α = β + 1 and Φ_s~α(E) = ∩_(β < α) Φ_s~β(E) for a measurable set E is contained in R. Then for each set A is contained in R its interior in T_s is given by the formula Int_s(A) = A ∩ Φ_s~β (B), where B is any measurable kernel of A and β is the smallest countable ordinal for which Φ_s~β (B) = Φ_s~(β+1)(B). Moreover, for each ordinal α, 1 ≤ α < ω_1, there exists a measurable set A such that Int_s(A) = A ∩ Φ_s~α (A) and lnt_s(A) ≠ A ∩ Φ_s~β(A) for β < α.