= Solution
The <vertex> stars form an <open> cover of $|K|$: every point has a nonempty positive-coordinate support, and therefore belongs to the star of at least one <vertex>. At a <vertex> $w$ itself, only its own barycentric coordinate is positive. Thus $w$ belongs to its own star and to no other <vertex> star.
Any finite selection of stars omits every <vertex> not among their finitely many labels. Infinitely many vertices therefore prevent a finite subcover, proving
$$
\boxed{|V|=\infty\Longrightarrow |K|\text{ is not compact}.}
$$
No local-finiteness hypothesis is needed for this <vertex-star cover obstruction to compactness>.
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