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

Asadzadeh, Mohammad Sina; Rezaei, Gholam Reza; Jamalzadeh, Javad

Khayyam J. Math. 7, No. 1, 77-85 (2021).

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.

MSC:

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

Keywords:

\( \epsilon \)-chainable; function-\(f\)-chainable; ultrapseudocompact

Citations:

Zbl 1010.54029

References:

[1]	A. Arhangel’skii and M. Tkachenko,Topological Groups and Related Structures, Atlantis Press, Paris, 2008. · Zbl 1323.22001
[2]	M. Atsuji,Uniform continuity of continuous functions of metric spaces, Pacific J. Math.8 (1958), no. 1, 11-16. · Zbl 0082.16207
[3]	G. Cantor,Über unendliche, lineare Punktmannigfaltigkeiten, Math. Ann.21(1883) 545- 591. · JFM 15.0452.03
[4]	P. Choudhary and K. Shrivastava,Locally function chainable sets in topological spaces, Int. J. Pure Appl. Math.102(2015), no. 1, 97-104.
[5]	M.I. Garrido and J.A. Jaramill,Lipschitz-type functions on metric spaces, J. Math. Anal. Appl.340(2008) 282-290. · Zbl 1139.46025
[6]	M.I. Garrido and A.S. Meroño,New types of completeness in metric spaces, Ann. Acad. Sci. Fenn. Math.39(2014) 733-758. · Zbl 1303.54010
[7]	S. Iliadis and V. Tzannes,Spaces on which every continuous map into a given space is constant, Canad. J. Math.38(1986), no. 6, 1281-1298. · Zbl 0599.54043
[8]	V. Iyer, K. Shrivastava and P. Choudhary,Chainability in topological spaces through continuous functions, Int. J. Pure Appl. Math.84(2013), no. 3, 269-277.
[9]	S. Kundu, M. Aggarwal and S. Hazra,Finitely chainable and totally bounded metric spaces: Equivalent characterizations, Topol. Appl.216(2017) 59-73. · Zbl 1355.54034
[10]	T. Nieminen,On ultrapseudocompact and related spaces, Ann. Acad. Sci. Fenn. Ser. A I Math.3(1977) 185-205. · Zbl 0396.54009
[11]	C.W. Patty,Foundations of Topology, Jones and Bartlett, Boston, 2009. · Zbl 1217.54001
[12]	K. Shrivastava and G. Agrawal,Characterization ofϵ-chainable sets in metric spaces, Indian J. Pure Appl. Math.33(2002), no. 6, 933-940. · Zbl 1010.54029
[13]	K. Shrivastava, P. Choudhary and V. Iyer,Self and strongly function chainable sets in topological spaces, Int. J. Pure Appl. Math.93(2014), no. 5, 765-773.