# Definition Given any [[set]] $X$ and $\mathcal{T} = \{U_i|i \in I\}$ is a certain collection of subsets of $X$. The pair $(X, \mathcal{T})$ is called a topological space if it obeys the following axioms: 1. $\emptyset, X \in \mathcal{T}$ 2. Any arbitrary (finite or infinite) union of members of $\mathcal{T}$ belongs to $\mathcal{T}$, i.e., given a (possibly infinite) subcollection $J$ of $I$, we have $\cup_{j\in J} U_j \in \mathcal{T}$ 3. Any finite intersection of members of $\mathcal{T}$ belongs to $\mathcal{T}$, i.e., given a subcollection $K$ of $I$, we have $\cap_{k\in K} U_k \in \mathcal{T}$ The members of $\mathcal{T}$ are called [[Open Set|open sets]], and $\mathcal{T}$ is said to endow $X$ with a topology.