Practical meaning of nowhere dense sets What is the practical meaning of a subset being nowhere dense on another set? What does it mean apart from the definition (its closure to have no interior points)?
 A: Historically, I think the notion arose as a natural opposite condition to that of being everywhere dense. Suppose that $D$ is everywhere dense in $\mathbb R$. Then $D$ is also dense in every open interval of $\mathbb R$. Conversely, if $D$ is dense in every open interval of $\mathbb R$, then $D$ is everywhere dense in $\mathbb R$. The converse is immediate because $\mathbb R$ itself is an open interval, but if this seems like cheating, note that the result still holds if we restrict ourselves to bounded open intervals. Thus, we can say that $D$ is everywhere dense in $\mathbb R$ if and only if $D$ is "everywhere locally dense in $\mathbb R$".
Therefore, a subset $E$ of $\mathbb R$ is not everywhere dense in $\mathbb R$ if and only if $E$ is not dense in some open interval (i.e. $E$ is not locally dense somewhere).
A really strong way for a subset $N$ of $\mathbb R$ to not be everywhere dense in $\mathbb R$ is the property of not being dense in every open interval (i.e. $N$ is not locally dense everywhere). This stronger way is exactly what being nowhere dense means.
It might help to think of these properties as describing sets that are everywhere big, somewhere big, and nowhere big. Or, in place of "big", you can use "thick".
