**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 τ_{1 }and τ_{2 }be two topologies on x we want to prove that τ_{1}∩ τ_{2 }(intersection of two topologies) is also topology on X.

- As τ
_{1 }and τ_{2 }are topologies on x

⇒φ, X ∈ τ_{1 }and φ, X ∈ τ_{2}

⇒φ, X ∈ τ_{1}∩ τ_{2 }

The first condition for topology is satisfied

- Let A
_{1}, A_{2}, A_{3}, —, A_{n}∈τ_{1}∩ τ_{2 }

We want to show that A_{1}∩ A_{2}∩ A_{3}∩ —∩ A_{n} ∈τ_{1}∩ τ_{2 }

A_{1}, A_{2}, A_{3}, —, A_{n} ∈τ_{1}∩ τ_{2 }

⇒ A_{1}, A_{2}, A_{3}, —, A_{n} ∈τ_{1 }and A_{1}, A_{2}, A_{3}, —, A_{n} ∈τ_{2}

As τ_{1} and τ_{2 }are topologies on X

A_{1}∩ A_{2}∩ A_{3}∩ —∩ A_{n} ∈τ_{1 }and A_{1}∩ A_{2}∩ A_{3}∩ —∩ A_{n} ∈τ_{2}

⇒ A_{1}∩ A_{2}∩ A_{3}∩ —∩ A_{n} ∈τ_{1}∩ τ_{2 }

The second condition for topology is satisfied

- Let A
_{1}, A_{2}, A_{3}, — ∈τ_{1}∩ τ_{2 }

We want to show that A_{1}∪ A_{2}∪ A_{3}∪ — ∈τ_{1}∩ τ_{2 }

A_{1}, A_{2}, A_{3}, — ∈τ_{1}∩ τ_{2 }

⇒ A_{1}, A_{2}, A_{3}, — ∈τ_{1 }and A_{1}, A_{2}, A_{3}, — ∈τ_{2}

As τ_{1} and τ_{2 }are topologies on X

A_{1}∪ A_{2}∪ A_{3}∪ — ∈τ_{1 }and A_{1}∩ A_{2}∩ A_{3}∩ — ∈τ_{2}

⇒ A_{1}∪ A_{2}∪ A_{3}∪ — ∈τ_{1}∩ τ_{2 }

The third condition for topology is satisfied

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

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