detials answer

We have
τπ = {U ⊂ Y : π−1(U) ⊂ τ}
Claim: This is topology.
1. As π is continuous, hence π−1Y ∈ τ and hence Y ∈ τπ. We have φ ∈ τπ.
2. Take any collection {Uα} ∈ τπ. Then we have π−1(Uα) ∈ τ Hence
π−1(∪αUα)τ Hence we have ∪αUα ∈ τπ
3. Take any finite collection {Un} ∈ τπ. Then we have π−1(Un) ∈...