CHARACTERIZATION OF ISOLATED POINTS IN T1 SPACES USING CHAINS

DURMISHI, Emin and MISAJLESKI, Zoran and RUSHITI, Agim and SADIKI, Flamure and IBRAIMI, Alit (2022) CHARACTERIZATION OF ISOLATED POINTS IN T1 SPACES USING CHAINS. Journal of Natural Sciences and Mathematics of UT, 7 (13-14). pp. 108-113. ISSN 2671-3039

Text JNSM 13-14 e formatuar-108-113.pdf Download (500kB)

Abstract

By the standard definition, a point x is an isolated point in a topological space if its corresponding one-element set is open. Here it is characterized the notion of isolated point using chains and open covers of the space. Namely, x is an isolated point in a T1 topological space if there exists an open covering of the space such that for any point of the space different from x, there is no chain in the covering joining it with the point x. Equivalently, it is provided a characterization of isolated point using the notion of pair of chain separated sets relatively a T1 space, while when using the notion of pair of weakly chain separated sets relatively a T1 space it is shown that only the sufficiency of the claim holds. Using these results, it is provided a characterization of discrete topological spaces and it is reproved that every compact Hausdorff space without isolated points is uncountable.

Item Type:	Article
Uncontrolled Keywords:	Isolated point, Chain connectedness, Chain separatedness, Open cover
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:	31 Oct 2022 12:52
Last Modified:	31 Oct 2022 12:52
URI:	http://eprints.unite.edu.mk/id/eprint/1065

Actions (login required)

View Item