2012/03/09 by Hüseyi̇n Çakallı, Huseyin Cakalli, Bipan Hazarika +2 · 1 citation
Decision Sciences · Mathematics · #Approximation Theory and Sequence Spaces #Fuzzy and Soft Set Theory #Rings, Modules, and Algebras #math.GM
paper · pdf · doi:10.48550/arxiv.1203.2003
16 pages. arXiv admin note: text overlap with arXiv:1005.4940
arxiv created 2012/03/09 · arxiv updated 2012/03/12
An ideal I is a family of subsets of positive integers N which is closed under taking finite unions and subsets of its elements. A sequence (xn) of real numbers is said to be I-convergent to a real number L, if for each ε> 0 the set \n:|xn-L|≥ ε\ belongs to I. We introduce I-ward compactness of a subset of R, the set of real numbers, and I-ward continuity of a real function in the senses that a subset E of R is I-ward compact if any sequence (xn) of points in E has an I-quasi-Cauchy subsequence, and a real function is I-ward continuous if it preserves I-quasi-Cauchy sequences where a sequence (xn) is called to be I-quasi-Cauchy when (Δxn) is I-convergent to 0. We obtain results related to I-ward continuity, I-ward compactness, ward continuity, ward compactness, ordinary compactness, ordinary continuity, δ-ward continuity, and slowly oscillating continuity.