Definition

Let $X=\overline{\mathbb{R}}$ be the Extended Real Line then $$\mathscr{T}=\left\{ A\subset X:\forall a \in A, \begin{cases} \exists\epsilon>0\ s.t. \ (a-\epsilon,a+\epsilon)\subseteq A&\text{if } a\in\mathbb{R} \\ \exists M>0\ s.t. \ (M,+\infty]\subseteq A&\text{if }a=+\infty \\ \exists M>0\ s.t. \ [-\infty,-M)\subseteq A&\text{if }a=-\infty \end{cases} \right\}$$is called the standard Topology on $\overline{\mathbb{R}}$.