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

摘要

It is a meaningful issue that under what condition neighborhoods induced by acovering are equal to the covering itself. A necessary and sufficient conditionfor this issue has been provided by some scholars. In this paper, through acounter-example, we firstly point out the necessary and sufficient condition isfalse. Second, we present a necessary and sufficient condition for this issue.Third, we concentrate on the inverse issue of computing neighborhoods by acovering, namely giving an arbitrary covering, whether or not there existsanother covering such that the neighborhoods induced by it is just the formercovering. We present a necessary and sufficient condition for this issue aswell. In a word, through the study on the two fundamental issues induced byneighborhoods, we have gained a deeper understanding of the relationshipbetween neighborhoods and the covering which induce the neighborhoods.