Topology

topology

Definition

Topology

A topology on a set $X$ is a collection $\tau$ of open sets satisfying

1. $\varnothing \in \tau$ and $X\in \tau$,
2. for every family $(U_i{)}_{i\in I}\subseteq \tau$, the union ${\bigcup }_{i\in I}{U}_i$ is in $\tau$,
3. for every finite sub-collection $U_1,\dots ,U_n\in \tau$, the intersection ${\bigcap }_{k=1}^n{U}_k$ is in $\tau$.