Let $\displaystyle E, F \in \mathbb{\epsilon}$, show that $\displaystyle E \cap F \in \tau$.

$\displaystyle \epsilon$ is a collection of subsets of a set $\displaystyle X$ that satisfies:
(1) the union of all members of $\displaystyle \epsilon$ is $\displaystyle X$, and
(2) $\displaystyle \forall E_1 \in \epsilon, \forall E_2 \in \epsilon, \forall x \in E_1 \cap E_2$, there is $\displaystyle U \in \epsilon$ such that $\displaystyle x \in U \subset E_1 \cap E_2$

$\displaystyle \epsilon$ is a base for $\displaystyle \tau$

$\displaystyle \tau = \{T \subset X: T $ is the union of a collection of members of $\displaystyle \epsilon\}$ is a topology on $\displaystyle X$