Abstract
An ideal $I$ is a family of subsets of positive integers $\mathbb{N}$ which is closed under taking finite unions and subsets of its elements. In this paper, we introduce the notion of ideally slowly oscillating sequences, which is lying between ideal convergent and ideal quasi-Cauchy sequences, and study on ideally slowly oscillating continuous functions, and ideally slowly oscillating compactness.
Highlights
An ideal I is a family of subsets of positive integers N which is closed under taking finite unions and subsets of its elements
We introduce the notion of ideally slowly oscillating sequences, which is lying between ideal convergent and ideal quasi-Cauchy sequences, and study on ideally slowly oscillating continuous functions, and ideally slowly oscillating compactness
We note that the definition of a quasi-Cauchy sequence is a special case of an ideal quasi-Cauchy sequences where I is taken as the finite subsets of the set of positive integers
Summary
An ideal I is a family of subsets of positive integers N which is closed under taking finite unions and subsets of its elements. Recall that a sequence x = (xn ) of points in R is said to be I-convergent to the number l if for every ε > 0, the set {n ∈ N : |xn − l| ≥ ε} ∈ I. In this case we write I-lim xn = l. We note that the definition of a quasi-Cauchy sequence is a special case of an ideal quasi-Cauchy sequences where I is taken as the finite subsets of the set of positive integers.
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