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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.

A contravariant set-valued functor on schemes is a Zariski sheaf if compatible local elements on any Zariski cover glue uniquely. Explicitly, $F(T)$ is the equalizer of ${\prod }_{i}F(U_i)\rightrightarrows {\prod }_{i,j}F(U_i{\times }_T{U}_j)$. The functor of points of every scheme is a Zariski sheaf because morphisms of schemes glue over open covers.