A topology on \( X \) is a collection \( \mathcal{T} \) of subsets of \( X \), that is: it is a subset of the \( P \left ( X \right ) \), with the following properties:

- \( \emptyset , X \in \mathcal{T} \)
- Suppose \( \left \lbrace U_{\alpha} \right \rbrace \) is a family of sets in \( \mathcal{T} \) then
- Suppose \( \left \lbrace U_{i} \right \rbrace_{i = 1}^{n} \) is a finite family of set in \( \mathcal{T} \) then