%0 Journal Article
%T Topological characterization of chainable sets and chainability via continuous functions
%J Khayyam Journal of Mathematics
%I Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with the Center of Excellence in Analysis on Algebraic Structures)
%Z 2423-4788
%A Rezaei, Gholam Reza
%A Asadzadeh, Mohammad Sina
%A Jamalzadeh, Javad
%D 2021
%\ 01/01/2021
%V 7
%N 1
%P 77-85
%! Topological characterization of chainable sets and chainability via continuous functions
%K ϵ-chainable
%K function-f-chainable
%K ultrapseudocompact
%R 10.22034/kjm.2020.219320.1710
%X 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.
%U https://www.kjm-math.org/article_123052_e4c5804fe16f5bd6826091dbe093035d.pdf