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)?

As an example demonstrating that two topologies need not be comparable, let $X = \{a, b\}$ (a set with just two elements). Define on $X$ the topologies
$$
\sigma = \{\emptyset, \{a\}, X\}
$$
and
$$
\tau = \{\emptyset, \{b\}, X\}.
$$
(the sets listed are taken to be the open sets of the topology). Both $(X, \sigma)$ and $(X, \tau)$ are topological spaces (you can check that they satisfy the axioms), but neither one is contained in the other.
