Soft ω-local indiscreetness as a weaker form of both soft local countability and is introduced. Then ω-regularity regularity defined investigated. Additionally, ω-T2 new topological property that lies strictly between T2 T1 It proved anti-local sufficient condition for equivalence ω-locally (resp. ω-regularity) locally ω-regularity). it the induced spaces indiscrete ω-regular, ω-T2) space are s...

For a family F⊆ωω we define the ideal I(F) on ω×ω to be generated by family{A⊆ω×ω:∃f∈F∀∞n(|{k:(n,k)∈A}|≤f(n))}. Using ideals of form I(F), show that structure Borel in-between two well known idealsED={A⊆ω×ω:∃m∀∞n(|{k:(n,k)∈A}|<m))} andFin⊗Fin={A⊆ω×ω:∀∞n(|{k:(n,k)∈A}|<ℵ0))} in Katětov order is fairly complicated. Namely, there copy P(ω)/Fin ED and Fin⊗Fin, consequently are increasing decreasing ...