Chain Connected Sets in a Topological Space

Z. Misajleski, N. Shekutkovski, A. Velkoska

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.