Cakalli, Huseyin2024-07-122024-07-1220110895-71771872-947910.1016/j.mcm.2011.04.0372-s2.0-79957867673https://dx.doi.org/10.1016/j.mcm.2011.04.037https://hdl.handle.net/20.500.12415/8326A subset E of a metric space (X, d) is totally bounded if and only if any sequence of points in E has a Cauchy subsequence. We call a sequence (x(n)) statistically quasi-Cauchy if st - lim(n ->infinity) d(x(n+1), x(n)) = 0, and lacunary statistically quasi-Cauchy if S-theta - lim(n ->infinity) d(x(n+1), x(n)) = 0. We prove that a subset E of a metric space is totally bounded if and only if any sequence of points in E has a subsequence which is any type of the following: statistically quasi-Cauchy, lacunary statistically quasi-Cauchy, quasi-Cauchy, and slowly oscillating. It turns out that a function defined on a connected subset E of a metric space is uniformly continuous if and only if it preserves either quasi-Cauchy sequences or slowly oscillating sequences of points in E. (C) 2011 Elsevier Ltd. All rights reserved.eninfo:eu-repo/semantics/openAccessQuasi-Cauchy sequencesSlowly oscillating sequencesSummabilityTotal boundednessUniform continuityArticle162405.JunN/A162054WOS:000291243300037Q1