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 dont 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?