Usual Topology

Definition

$\mathcal{U}=\left\{O|O={\bigcup }_{i=1}^{\infty }(a_i,b_i)\text{s.t.}(a_i,b_i\in \Omega \text{and}{a}_i\le b_i)\right\}$

$\mathcal{U}$ is a Family of Sets whose elements can be expressed as a countable union of open intervals.

Facts

The usual topological space $(\mathbb{R},\mathcal{U})$ satisfies $\forall o\in O,\exists a,b\in \mathbb{R},o\in (a,b)\subset O$ where $O$ is the set of open intervals