Topological Spaces

Remarks on topological space

Definition

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.