# analysis help

• January 19th 2009, 02:38 PM
dori1123
analysis help
Let $E, F \in \mathbb{\epsilon}$, show that $E \cap F \in \tau$.

Definition:
$\epsilon$ 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 $X$