# The intersection of two topologies on X is again a topology on x.( theorem solution )

#### Theorem

Let X be a non-empty finite or infinite set.

Prove that the intersection of two topologies on X is again a topology on x.

##### Proof

Let τ1 and τ2 be two topologies on x we want to prove that τ1∩ τ2 (intersection of two topologies) is also topology on X.

• As τ1 and τ2 are topologies on x

⇒φ, X ∈ τ1 and φ, X ∈ τ2

⇒φ, X ∈ τ1∩ τ2

The first condition for topology is satisfied

• Let A1, A2, A3, —, An ∈τ1∩ τ2

We want to show that A1∩ A2∩ A3∩ —∩ An ∈τ1∩ τ2

A1, A2, A3, —, An ∈τ1∩ τ2

⇒ A1, A2, A3, —, An ∈τ1 and A1, A2, A3, —, An ∈τ2

As τ1 and τ2 are topologies on X

A1∩ A2∩ A3∩ —∩ An ∈τ1 and A1∩ A2∩ A3∩ —∩ An ∈τ2

⇒ A1∩ A2∩ A3∩ —∩ An ∈τ1∩ τ

The second condition for topology is satisfied

• Let A1, A2, A3, — ∈τ1∩ τ2

We want to show that A1∪ A2∪ A3∪ — ∈τ1∩ τ2

A1, A2, A3, — ∈τ1∩ τ2

⇒ A1, A2, A3, — ∈τ1 and A1, A2, A3, — ∈τ2

As τ1 and τ2 are topologies on X

A1∪ A2∪ A3∪ — ∈τ1 and A1∩ A2∩ A3∩ — ∈τ2

⇒ A1∪ A2∪ A3∪ — ∈τ1∩ τ

The third condition for topology is satisfied

Thus, the intersection of two topologies on X is again a topology on X.