Cantor's Intersection Theorem

Definition

Cantor’s Intersection Theorem

Consider a Topological Space $(X,\mathcal{T})$ and a nested sequence $(E_n{)}_{n\in \mathbb{N}}\text{s.t.}\forall n\in \mathbb{N},E_{n+1}\subset E_n$ of non-empty Compact closed subsets of $X$. Then, $\bigcap _{n\in \mathbb{N}}{E}_n\ne \varnothing$

Nested Intervals Theorem

Consider a Sequence $(I_n{)}_{n\in \mathbb{N}}$, where $I_n=[a_n,b_n]$, of closed intervals. The sequence of intervals $(I_n{)}_{n\in \mathbb{N}}$ is called a sequence of nested intervals if $\forall n\in \mathbb{N},I_{n+1}\subset I_n\text{and}\underset{n\to \infty }{\lim }(b_n-a_n)=0$

If $(I_n{)}_{n\in \mathbb{N}}$ is a sequence of nested intervals in real numbers, then $\exists !x\in \mathbb{R},x\in \bigcap _{n\in \mathbb{N}}{I}_n$

It is the special case of the Cantor’s intersection theorem.