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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, provingNo local-finiteness hypothesis is needed for this vertex-star cover obstruction to compactness.