definition. A basis of a Topology on a set $X$ is a collection $\mathcal{B}$ of subsets of $X$ such that:

• for each $x \in X$ there is a $B \in \mathcal{B}$ such that $x \in B$
• given $B_1, B_2 \in \mathcal{B}$, then $x \in B_1 \cap B_2$, then there is $B_3 \in \mathcal{B}$ such that $x \in B_3 \subset B_1 \cap B_2$

motivation for definition . See definition of open set in the Standard topology in Real analysis. As you can see the definition of the open set is in terms of open balls, in precisely such a way that open balls are said to generate the standard topology! (see below for definition of topology generated by a basis)

Topology generated by a basis

Lemma to compare basis

Equivalent definition of basis, defined when the Topology is given.

Can also be defined in terms of Filter bases