Verified answer

Here's the definition of a topology τ on X (see link):

Let X be a set and let τ be a family of subsets of X. Then τ is called a topology on X if:

1. Both the empty set and X are elements of τ.

2. Any union of sets in τ is an element of τ.

3. Any intersection of finitely many elements of τ is an element of τ.

So the question amounts to verifying 1, 2, and 3 for the intersection ⋂[i∈I]T_i

of the topologies T_i.

1: Since each T_i is a topology on X, ∅ ∈ T_i and X ∈ T_i for all i∈I.

Conclude ∅ ∈ ⋂[i∈I]T_i and X ∈ ⋂[i∈I]T_i.

2: Let V = {v_j : j∈J} be any collection of sets, each v_j of which is in ⋂[i∈I]T_i.

(In other words, let V ⊆ ⋂[i∈I]T_i).

Then V ⊆ T_i for each i ∈ I. So, because T_i is a topology, the union ⋃[j∈J]v_j is in T_i.

Conclude ⋃[j∈J]v_j ∈ ⋂[i∈I]T_i.

3: Similar to 2. Now assume V = {v_j : j∈J} is a finite collection (i.e., J is finite),

with each v_j in ⋂[i∈I]T_i. Again, V ⊆ T_i for each i ∈ I. So, because T_i is a topology,

and J is finite, the intersection ⋂[j∈ j]v_j is in T_i.

Conclude ⋂[j∈J]v_j ∈ ⋂[i∈ I]T_i.