Proof That Union and Intersection of Subsets of a Set X Open in X Are a Topology for X

Theorem

Letrepresent a set anda family of subsets ofsuch that

1.

2. The union of any elements ofis a member of

3. The intersection of any elements ofis a member of

Then ifis a family of subsets ofsuch thatif and only ifthenis a topology on

Proof

Sinceand

Supposethen

Alsoso

This can be extended to any intersection of sets.

Supposeis a family of sets ofis a family of subsets ofand

Henceis a topology onElements ofare called open sets.