= Solution
The <quotient topology> on $\widetilde X$ is
$$
\mathcal T_{\widetilde X}
=\{U\subseteq\widetilde X:\pi^{-1}(U)\text{ is open in }X\}.
$$
Since inverse images preserve arbitrary unions and finite intersections,
$$
\pi^{-1}\!\left(\bigcup_\alpha U_\alpha\right)
=\bigcup_\alpha\pi^{-1}(U_\alpha),
\qquad
\pi^{-1}(U\cap V)=\pi^{-1}(U)\cap\pi^{-1}(V).
$$
Also $\pi^{-1}(\varnothing)=\varnothing$ and $\pi^{-1}(\widetilde X)=X$. The displayed collection therefore satisfies all the <topology axioms>.
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