= Vertex-star cover obstruction to compactness
The <topological realization> of a <simplicial complex> with infinitely many <vertices> is not <compact>. Its vertex <open star in a simplicial complex> form an <open cover>, but a vertex belongs only to its own star. Thus every finite collection of vertex stars omits all the other vertices. The argument uses the <weak topology of a simplicial realization>, in which a vertex star is <open> because its complement is the realization of the subcomplex of simplices missing that vertex. No local finiteness assumption is required.
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