A pair $(X,\tau )$ is called a generalized topological space if $\tau$ is collection of subsets of $X$ so that $\emptyset \in \tau$ and $\tau$ is closed under arbitrary unions. A subset $A$ of GTS $(X,\tau )$ is strongly nowhere dense if for any nonempty open set $U\in \tau$, there exists nonempty open set $V \subset U$ such that $V\cap A=\emptyset$.
I need an example of a generalized topological space $X$ (which is not topological space) contains a strongly nowhere dense subset?
I add another answer in light of your comment to the answer above. Here I construct a generalized topological space (which is not a topological space) along with a non-empty strongly nowhere dense subset.
Let $X = \{0,1,2,3\}$ and let $$\tau = \big\{\{0,1\}, \{1,2\}, \{0,1,2\}, X\big\}.$$
Note that $X$ is the only member of $\tau$ containing $3$. It is easy to see that $(X,\tau)$ is a generalized topological space, but it is not a topological space, because $\{0,1\}, \{1,2\}\in \tau$, but $\{1\} = \{0,1\}\cap \{1,2\} \notin \tau$.
Let $A = \{3\}$. It is easy to see that $A$ is strongly nowhere dense.
Let $(X,\tau)$ be any generalized topological space. If $A\subseteq X$ is open and $A\neq \emptyset$, it cannot be be strongly nowhere dense: Let $U=A$, then for every nonempty open set $V\subseteq U$ we have $V \cap A = V\cap U = V \neq \emptyset$.
So $A = \emptyset$ is the only strongly nowhere dense subset of $(X,\tau)$.
