Unions and ideals of locally strongly porous sets

Abstract

For subsets of R + = [0, ∞) we introduce a notion of coherently porous sets as the sets for which the upper limit in the definition of porosity at a point is attained along the same sequence. We prove that the union of two strongly porous at 0 sets is strongly porous if and only if these sets are coherently porous. This result leads to a characteristic property of the intersection of all maximal ideals contained in the family of strongly porous at 0 subsets of R + . It is also shown that the union of a set A ⊆ R + with arbitrary strongly porous at 0 set is porous at 0 if and only if A is lower porous at 0 .

A relation between porosity convergence and pretangent spaces

“…In [3] the definition of porosity which was given for the subsets of real numbers, have been redefined for the subsets of natural numbers by using a special function which is called scaling function.…”

“…Let µ : N → R + be a strictly decreasing function such that lim n→∞ µ(n) = 0 and let A be a subset of N. Now, let us recall from [3] that upper and lower porosity of A at infinity as follows pµ (A) := lim sup (1) ) − µ(n (2) )| : n n (1) < n (2) , (n (1) , n (2) ) ∩ A = ∅ .…”

“…From the definitions of upper and lower porosity of a subset of N at infinity, we have the following trivial result [3].…”

“…Let A and B be any subsets of R + . We shall write A B, if for every sequence (a n ) n∈N ∈ Ã { 0}, there is a sequence (t n ) n∈N ∈ B { 0}, such that lim n→∞ a n /t n = 1 holds [3].…”

“…Unlike all types of convergences given in the literature with different forms, porosity convergence as relatively new is defined in [2]. The basis of this study lies in redefinition of the porosity notion from a point in [0, ∞) to infinity in natural numbers [3].…”