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

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

\epsilon is a base for \tau

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