@article { author = {Rezaei, Gholam Reza and Asadzadeh, Mohammad Sina and Jamalzadeh, Javad}, title = {Topological characterization of chainable sets and chainability via continuous functions}, journal = {Khayyam Journal of Mathematics}, volume = {7}, number = {1}, pages = {77-85}, year = {2021}, publisher = {Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with the Center of Excellence in Analysis on Algebraic Structures)}, issn = {2423-4788}, eissn = {2423-4788}, doi = {10.22034/kjm.2020.219320.1710}, abstract = {In 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 their definition appropriately and consequently, we give appropriate results and counterexamples.}, keywords = {ϵ-chainable,function-f-chainable,ultrapseudocompact}, url = {https://www.kjm-math.org/article_123052.html}, eprint = {https://www.kjm-math.org/article_123052_e4c5804fe16f5bd6826091dbe093035d.pdf} }