One of the answers given here wrote
A subset $Y \subseteq X$ is called to be somewhere dense if there exists a non-empty open set $U\subseteq X$ such that we have $\overline{Y\cap U}=\overline{U}.$ As one can see, here by some where we actually mean an open set; This attitude seems quite natural since the open sets constitute actually the most fundamental part of a topological space.
A subset $Y \subseteq X$ is called nowhere dense, if it is not the case that it is somewhere dense. It is easy to see that $Y$ is nowhere dense if and only if $\overline{Y}$ does not contain a non-empty open set; the latter is equivalent to the standard definition of a nowhere dense set.
How does one go from (2) to (3)? Shouldn't the case of "not somewhere dense" be
For every open set $U\subset X$ we don't have $\overline{Y \cap U} = \overline{U}?$ Or since $\overline{Y \cap U} \subset \overline{Y} \cap \overline{U} \subset \overline{U}$, this is equivalent to never having $\overline{U} \subset \overline{Y\cap U}?$
Best Answer
If $Y$ is nowhere dense in the sense derived from 2., so not anywhere dense, consider $\overline{Y}$: if this contained a non-empty open set $U \subseteq \overline{Y}$, then $$\overline{U \cap Y} = \overline{U \cap \overline{Y}}\tag{1}$$ (as $U$ is open, this holds for any subset $Y$), and the latter equals $\overline{U}$, and so $Y$ would be dense in $U$, contradicting its definition.
If however we know that $Y$ is somewhere dense, say in $U$, we have
$$U \subseteq \overline{U}= \overline{U \cap Y} \subseteq \overline{Y}$$
and $\overline{Y}$ has non-empty interior. So the definitions are equivalent.