Topology

Definition: A topology on a set \(X\) is a collection \(\mathcal{T}\) of subsets of \(X\) having the following properties:

1. \(\varnothing \in \mathcal{T}\) and \(X \in \mathcal{T}\).
2. The union of the elements of any subcollection of \(\mathcal{T}\) is in \(\mathcal{T}\).
3. The intersection of the elements of any finite subcollection of \(\mathcal{T}\) is in \(\mathcal{T}\).

A set \(X\) for which a topology \(\mathcal{T}\) has been specified is called a topological space.