Let $X$ be a set and $\mathcal{T}$ a set of subsets of $X$. > [!NOTE] Definition > $\mathcal{T}$ is said to a topology on $X$ iff it satisfies the following > > (T1) $\emptyset, X\in \mathcal{T}$, > (T2) $\mathcal{T}$ is closed under arbitrary unions, > (T3) $\mathcal{T}$ is closed under finite intersection. > > $(X, \mathcal{T})$ is called a topological space and a subset of $X$ is called *open* only if it lies in $\mathcal{T}$.