Nowhere dense sets

Let \(X\) be a topological space and \(A\subseteq X\). The following are equivalent:

• \(A\) is nowhere dense in \(X\), that is, \(\qty{\overline{A}}^\circ = \emptyset\);

• every nonempty open set \(U\subseteq X\) contains a nonempty open set \(V\) with \(V\cap A = \emptyset\);

• \(A\) is not dense in any nonempty open subset of \(X\).

For \(X = \RR\), \(A\) is nowhere dense if and only if every open interval \(I\) contains an open interval \(S\subseteq I\) with \(S\cap A = \emptyset\), that is, \(\overline{A}\) contains no open interval.