Interior in the simple density topology

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 ⊂ R . Then for each set A ⊂ 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 Int s ( A ) ≠ A ∩ Φ s β ( A ) for β < α .