Topological characterization of chainable sets and chainability via continuous functions. (English) Zbl 1474.54084

Summary: In the last decade, the notions of function-\(f\)-\( \epsilon \)-chainability, uniformly function-\(f\)-\( \epsilon \)-chainability, function-\(f\)-\( \epsilon \)-chainable sets and locally function-\(f\)-chainable sets were studied in some papers. We show that the notions of function-\(f\)-\( \epsilon \)-chainability and uniformly function-\(f\)-\( \epsilon \)-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-\(f\)-\( \epsilon \)-chainable (resp., locally function-\(f\)-chainable). Further, we provide a criterion for connectedness with covers. In the paper [K. Shrivastava and G. Agrawal, Indian J. Pure Appl. Math. 33, No. 6, 933–940 (2002; Zbl 1010.54029)], 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.


54D99 Fairly general properties of topological spaces
54E35 Metric spaces, metrizability


Zbl 1010.54029


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