generated from smyalygames/quartz
12 lines
554 B
Markdown
12 lines
554 B
Markdown
# 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 $x_{i} = x, \; i \geq j$.
|
|
|