Topological Spaces
A topology on a set X is a collection \mathcal{T} of subsets of X having the following properties:
The empty set \varnothing and the set itself X are in \mathcal{T}.
Any union of elements in \mathcal{T} is in \mathcal{T}.
Any finite intersection of elements in \mathcal{T} is in \mathcal{T}.
A topological space is an ordered pair (X,\mathcal{T}) such that X is a set and \mathcal{T} is a topology on X.
Let \mathcal{T} be a topology on a set X. We say that a subset U of X is an open set if U is in \mathcal{T}.
Let \mathcal{T} and \mathcal{T}^\prime be topologies on a set X.
If \mathcal{T} \subset \mathcal{T}^\prime, we say that \mathcal{T}^\prime is finer than \mathcal{T}, and that \mathcal{T} is coarser than \mathcal{T}^\prime.
TO-DO: Explanation, examples, etc.