Thread: Topology Proof

If T₁ and T₂ are topologies for X and B={(G₁∩G₂)|G₁∈T₁, G₂∈T₂},
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 T₁ ⊆ T and T₂ ⊆ T}=>{(G₁∩G₂)|G₁∈T₁, G₂∈T₂} 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)

Originally Posted by Kiefer

If T₁ and T₂ are topologies for X and B={(G₁∩G₂)|G₁∈T₁, G₂∈T₂},
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*

I find your notation so hard to follow.
(3) If then such that ;
such that .

but