Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with the Center of Excellence in Analysis on Algebraic Structures)Khayyam Journal of Mathematics2423-47887120210101Topological characterization of chainable sets and chainability via continuous functions778512305210.22034/kjm.2020.219320.1710ENGholam Reza RezaeiDepartment of Mathematics, University of Sistan and Baluchestan, Zahedan, IranMohammad Sina AsadzadehDepartment of Mathematics, University of Sistan and Baluchestan, Zahedan,
Iran.Javad JamalzadehDepartment of Mathematics, University of Sistan and Baluchestan, Zahedan,
Iran.Journal Article20200209In the last decade, the notions of function-f-ϵ-chainability, uniformly function-f-ϵ-chainability, function-f-ϵ-chainable sets and locally functionf-chainable sets were studied in some papers. We show that the notions of function-f-ϵ-chainability and uniformly function-f-ϵ-chainability are equivalent to the notion of non-ultrapseudocompactness in topological spaces. Also, all of these are equivalent to the condition that each pair of non-empty subsets (resp., subsets with non-empty interiors) is function-f-ϵ-chainable (resp., locally function-f-chainable). Further, we provide a criterion for connectedness with covers. In the paper "Characterization of ϵ-chainable sets in metric spaces" (Indian J. Pure Appl. Math. 33 (2002), no. 6, 933{940), the chainability of a pair of subsets in a metric space has been defined wrongly and consequently Theorem 1 and Theorem 5 are found to be wrong. We rectify<br /> their definition appropriately and consequently, we give appropriate results and counterexamples.https://www.kjm-math.org/article_123052_e4c5804fe16f5bd6826091dbe093035d.pdf