Finite Intersection Property

Definition

Consider a Topological Space $(X,\mathcal{T})$ and a collection of subsets $\{U_i{\}}_{i\in I}\subset X$. The collection has the finite intersection property if for every finite subcollection of $\{U_i{\}}_{i\in I}$, the intersection is nonempty $\{U_i{\}}_{i\in I}\subset X\text{s.t.}(\forall J\subset I\text{s.t.}|J|<\infty ,{\bigcap }_{j\in J}{U}_j\ne \varnothing )$

Examples

Consider a Standard Topology on real numbers $(\mathbb{R},\mathcal{U})$ and a collection of subsets $\{U_i{\}}_{i\in I}\subset X$ and a collection of subsets $\{(0,\frac{1}{n}]{\}}_{n\in \mathbb{N}}$. Then, the collection have finite intersection property.