# Definition Given $F \subset \wp(X)$. The **weakest topology** on $X$ that contains $F$ is the intersection of all the [[Topology|topologies]] that contains $F$. This is a [[Topology|topology]], and consists of $\emptyset$, $X$, and all unions of finite intersections of members from $F$. > [!example] > $F \subset \tau$ > $\textvisiblespace \cap \tau \ni U_{i} \implies U_{i} \in \tau$ > $\implies \cap_{i \in F} \; U_{i} \in \tau \implies \cap U_{i} \in \cap_{F \in \tau} \;\tau$ > > $x_{i} \to x$ > $\exists j$ such that $x_{i} = x, \; i \geq j$.