Definition 1

A subset $A$ of a topological space $X$ is said to be nowhere dense if its closure has no interior points.

Equivalently, $A$ is nowhere dense in $X$ if and only if the exterior of $A$ is dense in $X$.

A closed set $A$ is nowhere dense if and only if it has no interior points; that is, if and only if it coincides with its frontier. An arbitrary subset $A$ is nowhere dense if and only if the closure of $A$ is nowhere dense. Every subset of a nowhere dense set is nowhere dense.