Neighborhood

Let $(X, \tau)$ - a topological-space.

An open-set $$U(x) \in \tau$$ is a neighborhood of a dot $$x \in X$$ if $$x \in U(x)$$.

Definitive family of neighborhoods

A family of neighborhoods ${U_{\alpha}(x)}$ considered definitive if $$ \forall U(x) \in \tau: \exists U_{\alpha}(x) \hookrightarrow U_{\alpha}(x) \subset U(x). $$