Are open sets required to be in Topology T over set X? Assume that we choose two sets A and B to be open sets, contained in topology (X,T). A and B and the constructed sets that fulfill the requirements to make A and B open (unions and intersects of A, B, empty, and O) are represented by (X,O). Do all the constructed sets in (X,O) (sets other than A and B within (X,O)) have to be contained in the topology (X,T)?

If (X,O) is a topology which includes open sets A and B and constructed sets, then all (X,O) is contained in (X,T). Therefore (X,O) is a subset of (X,T).
Prove (X,O) is contained in (X,T):
If two sets A and B are considered open and contained in (X,T), all their constructed sets, which are required for A and B to be open, would also be contained in (X,T). Since (X,T) contains A and B, all unions and intersects of A, B, empty, and X are in (X,T). This is because the requirements for a set to be open are identical to the requirements for a set to be in a topology.
