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