What is the 'meaning' of nowhere dense set? In some books, nowhere dense set is defined to be $int(\bar A)=\emptyset$ but meanwhile is defined to be $int(A)=\emptyset$ in some books(e.g. Munkres).
So what is the 'meaning' (i.e motivation, intuitive/geometric meaning etc.) of nowhere dense set? Thank you.

You can think of the word "nowhere" as meaning "in no open set". So a subset $A \subset X$ is nowhere dense if it is not dense in any open set, or more precisely: for each open subset $U \subset X$ the set $A \cap U$ is not a dense subset of $U$ (with respect to the subspace topology on $U$).
