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

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.