Geodesic
Chain Connected Sets in a Topological Space
Kragujevac Journal of Mathematics, Tome 43 (2019) no. 4, p. 575

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Shekutkovski's paper \cite{ex2} compares two definitions of connectedness: the standard one and a definition using coverings. The second definition seems to be an effective description of quasicomponents. In our paper rather than as a space, we generalize the notion of connectedness as a set in a topological space called chain connected set. We also introduce a notion of two chain separated sets in a space and using this notion of chain, we study the properties of chain connected and chain separated sets in a topological space. Moreover, we prove the properties of connected spaces using chain connectedness. Chain connectedness of two points in a topological space is an equivalence relation. Chain connected components of a set in a topological space are a union of quasicomponents of the set, and if the set agrees with the space, chain connected components match with quasicomponents.
Classification : 54D05
Keywords: General topology, coverings, connectedness, chain connectedness, functionally separatedness, quasicomponents 
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     author = {Z. Misajleski and N. Shekutkovski and A. Velkoska},
     title = {Chain {Connected} {Sets} in a {Topological} {Space}},
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Z. Misajleski; N. Shekutkovski; A. Velkoska. Chain Connected Sets in a Topological Space. Kragujevac Journal of Mathematics, Tome 43 (2019) no. 4, p. 575 . http://geodesic.mathdoc.fr/item/KJM_2019_43_4_a5/