Condition for neighborhoods induced by a covering to be equal to the covering itself

TL;DR

This paper investigates the conditions under which neighborhoods induced by a covering are equal to the covering itself, correcting previous assumptions and providing new necessary and sufficient conditions for both direct and inverse problems.

Contribution

It corrects a false assumption in prior work and establishes new necessary and sufficient conditions for neighborhoods to match coverings and vice versa.

Findings

Counter-example disproves previous necessary and sufficient condition.

New necessary and sufficient conditions are established for neighborhoods equal to coverings.

Deeper understanding of the relationship between neighborhoods and coverings is achieved.

Abstract

It is a meaningful issue that under what condition neighborhoods induced by a covering are equal to the covering itself. A necessary and sufficient condition for this issue has been provided by some scholars. In this paper, through a counter-example, we firstly point out the necessary and sufficient condition is false. Second, we present a necessary and sufficient condition for this issue. Third, we concentrate on the inverse issue of computing neighborhoods by a covering, namely giving an arbitrary covering, whether or not there exists another covering such that the neighborhoods induced by it is just the former covering. We present a necessary and sufficient condition for this issue as well. In a word, through the study on the two fundamental issues induced by neighborhoods, we have gained a deeper understanding of the relationship between neighborhoods and the covering which induce the neighborhoods.