Consider the $2$ definitions:
A set $A$ in a topological space $(X,\tau)$ is said to be a nowhere dense set if it is not dense in any nonempty open set.
A Set $A$ in a topological space $(X,\tau)$ is said to be a nowhere dense set if $(\bar A)^0=\phi$.
I understand the two definitions as statements and know they are equivalent.
Suppose that $(\bar A)^0=\phi$,now take any open ball $V$ in $X$.Let if possible,for each open ball $U$ in $V$,$U\cap A\neq \phi$.Now for each of the open balls in $V$,choose an element of $A$ and consider the collection as $A_0$.Now for any point$v$ in $V$,every neighborhood of $v$ contains a point of $A_0$ by its construction.So,$V\subset \bar A_0 \subset \bar A$,i.e. $\bar A$ contains an open set which contradicts $(\bar A)^0=\phi$.
Conversely,suppose for each open ball $V$ in $X$,there exists $U$ open such that $U\subset V$ and $U\cap A=\phi$.To show,$\bar A$ can contain no open set.If possible $\bar A$ contains an open ball $V$,then for any open ball $U$ in $V$,if we take a point $x\in U$ then $x\in \bar A$,so $x$ is an adherent point of $A$ and since $U\in \eta_x$,so,$U\cap A\neq \phi$.
But I have not yet found the reason behind its name 'nowhere dense' i.e. I cannot feel it properly.I am looking for some diagram that would build my intuition on nowhere dense set.I have studied examples like Cantor set,but yet I am feeling uncomfortable with the notion of nowhere dense sets and why it means that points are not clustered very tightly in topological sense?I also want to know what is the motivation behind defining nowhere dense sets.
The above picture shows that for every open set $V$(yellow) in $(X,d)$,there is an open set $U\subset V$ (shown in white) which does not intersect $A$.
Some other constructions of nowhere dense sets are Cantor like construction,it forces the resulting set to be nowhere dense by deleting some open ball from each of the open balls of the whole set.For example Cantor set,Smith-Volterra Cantor set,Cantor dust in $\mathbb R^n$ and Cantor Cirlcle,Fractals-of-Cirlces that ca be found here.
Best Answer
This is based on an old (early 20th century) usage (gone mostly out of style) to call a set $A$ "somewhere $P$" if there is some non-empty open set $O$ such that $O \cap A$ has property $P$ in $O$ (somewhere = 'inside some non-trival open set') and a set that is "not somewhere dense", was called "nowhere dense" for short. This only explains the origin of this specific name. The definition is useful because of its connection to Baire's theorem and meagre sets.