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∩ τ2
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∩ τ2
The third condition for topology is satisfied
Thus, the intersection of two topologies on X is again a topology on X.
Add a Comment