Show that the intersection of two topologies on X is a topology.

Proof;

To show that the intersection of two topologies on X is a topology. OR Let T1 and T2 be topologies on X. Then T1∩T2 is a topology on X.

Related Theorems;

Show finite complement topology is, in fact, a topology.

If A is a subset of a topological space, then ∂(A)⊆Cl(A).

We have to prove that T1∩T2 satisfied the definition of topology on X.

1) The Union of any number of members of T1∩T2 belong to T1∩T2.

Let A1, A2, A3,..., An,... ∈ T1∩T2

⇒       A1, A2, A3,..., An,... ∈ T1 and A1, A2, A3,..., An,... ∈ T2

Since T1 and T2 are topologies on X,

⇒       A1 ∪ A2 ∪ A3 ∪,...,∪ An ∪,... ∈ T1 and A1 ∪ A2 ∪ A3 ∪,...,∪ An ∪,... ∈ T2

⇒      A1 ∪ A2 ∪ A3 ∪,...,∪ An ∪,... ∈ T1 ∩ T2

2) The intersection of finite number of members of T1∩T2 belong to T1∩T2.

Let    A1, A2, A3,..., An ∈ T1∩T2

⇒     A1, A2, A3,..., An ∈ T1 and A1, A2, A3,..., An ∈ T2

Since T1 and T2 are topologies on X,

⇒    A1 ∩ A2 ∩ A3 ∩,...,∩ An ∈ T1 and A1 ∩ A1 ∩ A3 ∩,...,∩ An ∈ T2

⇒   A1 ∩ A2 ∩ A3 ∩,...,∩ An ∈ T1∩T2

3) The empty set and set X belong to T1 ∩ T2.

Since T1 and T2 are topologies on X,

⇒          X, Φ ∈ T1 and X, Φ ∈ T2

⇒          X, Φ ∈ T1 ∩ T2

Hence proved that the intersection of two topologies on X is a topology.