ABSTRACT. This paper presents a density type topology with respect to an extension of Lebesgue measure involving sequence of intervals tending to zero. Some properties of such topologies are investigated.Let R denote a set of real numbers, N a set of natural numbers, and λ a Lebesgue measure on R. By L we understand a family of Lebesgue measurable sets, by L a family of Lebesgue measurable null sets, and by |I| a length of an interval I. By μ we denote any complete extension of Lebesgue measure λ, by S μ a domain of function μ, and by I μ a family of μ-null sets. If A, B are families of subsets of the space X, then we use notation A B = {C ⊂ X : C = A \ B, A ∈ A, B ∈ B} and AΔB = {C ⊂ X : C = AΔB, A ∈ A, B ∈ B}, where Δ is an operation of the symmetric difference. It is well-known that if A is σ-algebra of sets in X and B is σ-ideal of sets in X, then the family AΔB is the smallest σ-algebra containing A ∪ B.It is clear that x 0 ∈ R is a density point of a set A ∈ L ifIt is equivalent toThe above condition can be written as in [8]:where {J n } n∈N is a sequence of closed intervals.