Portion

of a set

An intersection of the set with an interval in the case of a set on a line, and with an open ball, an open rectangle or an open parallelopipedon in the case of a set in an $n$- dimensional space $(n\ge 2)$. The importance of this concept is based on the following. A set $A$ is everywhere dense in a set $B$ if every non-empty portion of $B$ contains a point of $A$, in other words, if the closure $\bar{A}\supset B$. The set $A$ is nowhere dense in $B$ if $A$ is nowhere dense in any portion of $B$, i.e. if there does not exist a portion of $B$ contained in $\bar{A}$.