Topology base

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

Subset $\beta \subset \tau$ called a topology base if each open-set can be decomposed into a (countable) union of elements of $\beta$.

Topology pre-base

Subset $\sigma \subset \tau$ called a topology pre-base if each element of base can be decomposed into a finite intersection of elements of $\sigma$.

Base criteria