I want to prove the following:
Let $f :\mathbb{S}^{n-1} \to X$ be continuous and surjective and let $\pi : D^n + X \to X\cup_f D^n$ be the canonical projection. Show that $\pi\mid_{D^n}: D^n \to X \cup_f D^n$ is a quotient map.
My guess is that since $f$ is continuous, we can invoke the universal property of quotient spaces to create a continuous function $\tilde{f} : X\cup_f D^n \to X$ such that $\tilde{f} = q \circ f$ where $q=\pi\mid_{D^n} = \pi \circ i$ and $i : D^n \to D^n + X$ is the inclusion map, but I'm been unable to complete the proof. I need to show that $q$ is surjective and that $q^{-1}(U)$ is open if and only if $U$ is open in $X\cup_f D^n$. One implication of this last part is done since both $\pi$ and $i$ are continuous, so it's the other implication where I need some help. Thank you in advance.