topology generated by basis

Overview

Given a topological basis, we can “generate” a topology for that basis.

Definition

For a set X and a basis , a set is an open set in the topology generated by the basis (call it ) if

Note that the important part is that . This is the condition that is ‘hard’, or not generally automatic.

Proof that is a topology

1. and X come automatically.
2. Arbitrary union: let be an arbitrary union of open sets (i.e. each follow above conditions) then so
3. Finite intersection: For open sets, so thus and . Then by the 2nd condition of the definition of being a basis,