Let $X = \omega = \{0,1,2,\ldots\}$ and let $$\tau = \big\{\{1,2\}, \{2, 3\}\big\} \cup \big\{S\subseteq \mathbb{N}: \{1,2,3\}\subseteq S\big\}.$$
Then $(X,\tau)$ is a generalized topological space (but not a topological space, since $\{2\}=\{1,2\}\cap\{2,3\}$ is not a member of $\tau$); moreover $\{0\}$ is strongly nowhere dense.