1. The problem asks for the definition of a neighborhood system. 2. In topology, a neighborhood system (or neighborhood filter) of a point $x$ in a topological space is the collection of all neighborhoods of $x$. 3. A neighborhood of $x$ is any set that includes an open set containing $x$. 4. Formally, the neighborhood system $\mathcal{N}(x)$ is defined as: $$\mathcal{N}(x) = \{N \subseteq X : \exists \text{ open } U \text{ with } x \in U \subseteq N\}$$ 5. This means every neighborhood contains an open set that contains the point $x$. 6. Neighborhood systems help describe local properties of spaces and are fundamental in defining continuity, convergence, and other topological concepts.