If T_{1}and T_{2}are topologies for X and B={(G_{1}∩G_{2})|G_{1}∈T_{1}, G_{2}∈T_{2}},

Then T*={G|For all p, if p∈G, then A∈B such that p∈A⊆G} is a topology.

So I need to show

(1) T*⊆P(X)

(2) ∅∈T* and X∈T*

(3) if A,B∈T*, then A∩B∈T*

(4) if A⊆T*, then ∪A∈T*

This is part of a problem proving T~=∩{T|T is a topology for X such that T_{1}⊆ T and T_{2}⊆ T}=>{(G_{1}∩G_{2})|G_{1}∈T_{1}, G_{2}∈T_{2}} is a base for T*...So is G an element of T~ (not sure what else it would be an element of)?

(I know the first two are kind of trivial, but if you are able to help me prove them anyways I would appreciate it)