Is there a way to characterize all nowhere dense sets of $\mathbb{R}$? I'm reading Functional Analysis from Joseph Muscat's book. Muscat asks the following question in an exercise:

What are the nowhere dense sets in $\mathbb{R}$?

He gives the hint that every open set in $\mathbb{R}$ is a countable union of disjoint open intervals.
By definition, closure of nowhere dense sets cannot contain a open ball which is an open interval in $\mathbb{R}$. Now, how can I use the hint further characterize nowhere dense sets in $\mathbb{R}$?
I do not need a complete answer. A nudge in the right direction will be appreciated!

The question as given in the book:

 A: Frankly, this is bizarre to me. The reference to the prior exercise showing that open sets in $\mathbb{R}$ are countable disjoint unions open intervals suggests that the desired classification is the following:

In $\mathbb{R}$, $A$ is nowhere dense iff $\overline{A}$ does not contain any nonempty open interval.

This is a trivial application of the prior exercise. However, the author has marked this exercise as particularly difficult (the "$*$" preceding it). Moreover, the "result" above doesn't even use the interesting part of the prior exercise! All it uses is that every open set contains a nonempty open interval, which is basically trivial. So this feels wrong. On the other hand, I see no other interpretation of the problem which makes sense.

For what it's worth, in my opinion there is no good classification of nowhere dense sets in $\mathbb{R}$. There are simply too many of them: since every subset of a nowhere dense set is nowhere dense, and there are nowhere dense sets of size continuum (e.g. the Cantor set), the number of nowhere dense subsets of $\mathbb{R}$ is as large as it could be: $2^{2^{\aleph_0}}$. By contrast, there are only $2^{\aleph_0}$-many open sets in $\mathbb{R}$. (This is a good exercise; the first step is to show that every open set in $\mathbb{R}$ can be written as a not-necessarily-disjoint union of countably many open intervals with rational endpoints, and that's worth doing even if you're not interested in cardinality.)
Moreover, nowhere dense sets need not be small in all senses. In particular, there are nowhere dense sets which have positive Lebesgue measure (e.g. the "fat Cantor sets"). In general, in contrast with null sets nowhere dense sets are not "ignorable" in analysis.
