Topology proof: smallest topology and base for topology

Thm: If Τ_{1 }and Τ_{2} are topologies for X and Τ*=∩{ Τ | Τ is a topology for X such that Τ_{1} ⊆ Τ, Τ_{2} ⊆ Τ},

then

(1) Τ* is the smallest topology for X containing both Τ_{1 }and Τ_{2}; and

(2) {(G_{1}∩G_{2})| G_{1} ∈ Τ_{1}, G_{2} ∈ Τ_{2}} is a base for Τ*

We know:

B is a base for Τ iff

(1) B⊆Τ and,

(2) For every G∈Τ, if p∈G, then there is B∈B such that p∈B⊆G.

Re: Topology proof: smallest topology and base for topology

Re: Topology proof: smallest topology and base for topology

For part one, I can prove that T* is a topology but am not sure how to prove it is the smallest one in X containing T1 & T2

Re: Topology proof: smallest topology and base for topology

Quote:

Originally Posted by

**Kiefer** For part one, I can prove that T* is a topology but am not sure how to prove it is the smallest one in X containing T1 & T2

Why is that?

Say is smaller.

But haven't you already incuded in