Proof That Union and Intersection of Subsets of a Set X Open in X Are a Topology for X
Letrepresent a set anda family of subsets ofsuch that
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
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.