Proposition 1

The union of a finite number of nowhere dense sets is nowhere dense.

It is enough to show that the union of two nowhere dense sets $A, B$ is nowhere dense, and without loss of generality we may assume that $A$ and $B$ are closed. The proposition is then equivalent to saying that the intersection of two dense open sets $\overline{C}A, \overline{C}B$ is dense. Now if $U$ is a non-empty open set, then $U \cap \overline{C}A$ is open and non-empty, hence

is open and non-empty.

Let $Y$ be a subspace of a topological space $X$. A subspace $A$ of $Y$ is said to be nowhere dense relative to $Y$ if $A$ is nowhere dense when considered as a subset of the topological space $Y$.