In Introductory Real Analysis by Kolmogorov and Fomin the following definitions are given:
Let A and B be two subsets of a metric space $R$. Then $A$ is said to be dense in $B$ if $B \subset [A]$. In particular, $A$ is said to be everywhere dense (in $R$) if $[A] = R$. A set $A$ is said to be nowhere dense if it is dense in no (open) sphere at all.
Here my question. If $A$ is dense in $B$, should not be $[A]=B$? Further, it is not clear to me the definition of nowhere dense.