I'm trying to prove the following: Let $x$ and $Y$ be locally compact Hausdorff spaces and $X^{*}$ and $Y^{*}$ be their one-point compactifications. Prove there is a homeomorphism $$(X \amalg Y)^{*} \cong X^{*} \vee Y^{*}$$
My attempt: Obviously the homeomorphism needs to be the map $f$ such that $(x,0) \mapsto (x,0)$, $(y,1) \mapsto (y,1)$, and $\infty \mapsto \{\infty_{X},\infty_{Y}\}$. This immeditly makes $f$ bijective. My problem is proving continuity of $f$. Notice $U$ is open in $X^{*} \vee Y^{*}$ if and only if $p^{-1}(U)$ is open where $p:X^{*} \amalg Y^{*} \rightarrow X^{*} \vee Y^{*}$ is the quotient map. Now $(X \amalg Y)^{*}$ and $X^{*} \amalg Y^{*}$ are clearly different spaces topologically, so I $p^{-1}(U)$ gives me no useful information to prove that $f^{-1}(U)$ is open. How could I resolve this issue?