The following is an exercise in Tommo Tom Dieck's book "Algebriac Topology"
If $X$ is a topological space, $A$ is a compact subspace of $X$ ans $p:X\to X/A$ is the canonical quotient map then for any topological space $Y$ the product map $p\times id_Y$ is a quotient map.
I need to prove It in order to demonstrate that $\Sigma(Y)$, the reduced suspension, is the same than the smash product of $S^1$ and $Y$.
What I have done is try to prove that if $U$ is a subset such that $(p\times id_Y)^{-1}$ is open, i.e., is in the form $\bigcup_{i\in\alpha}O_i\times B_i$ then $A\subset \bigcup_{i\in\alpha}O_i$, ando knowing that $A$ is compact this can be reduced yo a finite union.
I don't know how to proceed for now on. It would be very useful if you could give me a hint.