CHAIN OF A SET IN A COVERING AND CHAIN COMPONENTS UP TO A COVERING

DURMISHI, Emin and MISAJLESKI, Zoran and SADIKU, Flamure and IBRAIMI, Alit (2023) CHAIN OF A SET IN A COVERING AND CHAIN COMPONENTS UP TO A COVERING. Journal of Natural Sciences and Mathematics of UT, 8 (15-16). pp. 365-368. ISSN 2545-4072

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Abstract

A chain in the open covering V of a topological space X that joins U V and V V is a finite sequence of elements of V such that U is the first member, V is the last member and every two consecutive members of the sequence have a nonempty intersection. By chainV,V V it is meant the union of all elements of the covering for which there are chains joining them with V and chainV is the set that consists of all sets chainV for each V V. A chain in V that joins x X and y X is a finite sequence of elements of V such that x is contained in the first element of the sequence, y is contained in the last element and every two consecutive elements of the sequence have a nonempty intersection. A V -chain component of an element x X  , Ch x( , ), V is the set that consists of all y X  such that there exists a chain in V that joins x and y . We prove that chainV Ch x  ( , ) V for any V V and any x V , hence chainV consists of V -chain components. As a consequence, chain connectedness is characterized using the chainV notion.

Item Type:	Article
Uncontrolled Keywords:	Chain, Star of a set, Open covers, Chain connectedness.
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Engineering, Science and Mathematics > School of Mathematics
Depositing User:	Unnamed user with email zshi@unite.edu.mk
Date Deposited:	05 Nov 2023 00:09
Last Modified:	05 Nov 2023 00:09
URI:	http://eprints.unite.edu.mk/id/eprint/1538

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