Why is p × i a quotient map when p is quotient and Z is locally compact Hausdorff?

[ak416]

Prove: If p:X->Y is a quotient map and if Z is a locally compact Hausdorff space, then the map m: p x i : X x Z -> Y x Z is a quotient map.

Note: i is the identity map on Z i assume. There is a few lines of hints talking about using the tube lemma and saturated neighborhoods which i don't feel like writing at the moment (see Munkres p.186). The main problem i have have with this is why not say: p x i (U x V) = p(U) x i(V), so (p x i)^-1 (U x V) = p^-1(U) x i^-1(V) (is this not true?). Because if it is true then the result seems trivial since p is a quotient map and the identity does nothing to change the openness of V. What am i missing?

 

[AKG]

What does it mean for p x i to be a quotient map? You can use what you've done to prove that p x i is continuous, but you would need to do more to prove that it's a quotient map.