Let \(X\) be a topological space. A subset \(S\subseteq X\) is nowhere dense in \(X\) if its closure \(\overline{S}\) has empty interior, that is, no nonempty open subset of \(X\) is contained in \(\overline{S}\).
Equivalently, every nonempty open set \(U\subseteq X\) contains a nonempty open set \(V\) with \(V\cap S=\emptyset\), so \(S\) is not dense in any nonempty open subset of \(X\). Indeed, if \(\overline{S}\) has empty interior then \(V\coloneqq U\setminus\overline{S}\) is nonempty and open; conversely, a nonempty open \(W\subseteq\overline{S}\) contains no nonempty open set disjoint from \(S\). For \(X=\RR\), \(S\) is nowhere dense if and only if every open interval contains an open subinterval disjoint from \(S\).