= Solution
First prove the suggested density fact. If $Z$ is <dense> in $Y$, $V$ is <open> and $y\in V$, then every <open> <neighborhood> $W$ of $y$ has $W\cap V$ nonempty <open>. It meets $Z$, so $W$ meets $V\cap Z$. Hence
$$
V\subseteq\overline{V\cap Z}^{\,Y}.
$$
Identify $X$ with its <dense> embedded image in the <hausdorff compactification>. For $x\in X$, local compactness supplies an <open> <neighborhood> $U$ in $X$ and a <compact> <set> $K\subseteq X$ with $x\in U\subseteq K$. Choose an <open set> $V$ in $Y$ with $V\cap X=U$. The inclusion into $Y$ is <continuous>, so $K$ is <compact> in $Y$ and therefore <closed>, since $Y$ is <Hausdorff>. The density fact now yields
$$
x\in V\subseteq\overline{V\cap X}^{\,Y}=\overline U^{\,Y}\subseteq K\subseteq X.
$$
Thus every point of the embedded $X$ has a $Y$-open <neighborhood> contained in $X$. Consequently
$$
\boxed{h[X]\text{ is open in }Y.}
$$
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