= Solution
Use the <Stone-Čech extension property>: a discrete <topological space> $X$ embeds densely into a <compact> <Hausdorff space> $\beta X$, and every <continuous map> from $X$ to a <compact> <Hausdorff space> extends continuously to $\beta X$. Apply it to the <characteristic function> $\chi_A:X\to\{0,1\}$ of an arbitrary <subset> $A$; discreteness makes it <continuous>. Its extension $\widetilde\chi_A$ has <clopen> fibers, and
$$
\overline A^{\,\beta X}=\widetilde\chi_A^{-1}(1).
$$
Indeed that fiber is <closed> and contains $A$, while its openness and density of $X$ show that every point in the fiber is approached by points of $A$.
If $A$ and $B$ are disjoint, the extension of $\chi_A$ is one on $\overline A$ and zero on $\overline B$, so these <closures> are disjoint. The inclusion $X\to\beta X$ therefore satisfies the criterion from part (b), and its ultrafilter-limit map is both <surjective> and <injective>. Thus
$$
\boxed{\Upsilon(X)\longleftrightarrow\beta X,\qquad
\mathcal H\longmapsto\text{the unique limit of its image filter}.}
$$
This is a <bijection> of underlying <sets>; identifying the compatible <topology> on the <ultrafilter> <topological space> would give the familiar stronger <homeomorphism> statement, but is not needed for the requested conclusion.
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