Topological Space

A topological space is a pair $(X,\mathcal{T})$ consisting of a set $X$ and a collection $\mathcal{T}$ of subsets of $X$ such that the following conditions hold.

1. The empty set $\varnothing$ and the whole set $X$ belong to $\mathcal{T}$.

2. For every set $A$ and every family of subsets of $X$ $(U_a)_{a\in A}$ such that $U_a\in\mathcal{T}$ for every $a\in A$, the union

also belongs to $\mathcal{T}$.

3. For every natural number $n\in\mathbb{N}$ and every subsets $U_1,\dots,U_n\in\mathcal{T}$, the intersection

also belongs to $\mathcal{T}$.

The elements of $\mathcal{T}$ are called open sets, and $\mathcal{T}$ is called a topology on $X$.