覆盖诱导的邻域等于覆盖本身的条件
Condition for neighborhoods induced by a covering to be equal to the covering itself
摘要：
在什么条件下由覆盖诱导的邻域等于覆盖本身是一个有意义的问题。一些学者已经给出了这个问题的充分必要条件。本文通过一个反例，首先指出了该充要条件是不成立的。其次，我们给出了这个问题的一个充分必要条件。第三，我们专注于由覆盖计算邻域的逆问题，即给定一个任意覆盖，是否存在另一个覆盖使得由它导出的邻域正好是前一个覆盖。我们也给出了这个问题的一个充分必要条件。总之，通过对邻域诱导的两个基本问题的研究，我们对邻域与诱导邻域的覆盖之间的关系有了更深的认识。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
