Weakest Topology

Definition

Given $F\subset \wp (X)$. The weakest topology on $X$ that contains $F$ is the intersection of all the topologies that contains $F$. This is a topology, and consists of $\varnothing$, $X$, and all unions of finite intersections of members from $F$.

Example

$F\subset \tau$
$\text{textvisiblespace}\cap \tau \ni U_i⟹U_i\in \tau$
$⟹{\cap }_{i\in F}{U}_i\in \tau ⟹\cap U_i\in {\cap }_{F\in \tau }\tau$

$x_i\to x$
$\exists j$ such that $x_i=x,i\ge j$.