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.