Past exam of the topology course of the University of Oxford / 2014 / 3 / a

An abstract simplicial complex $K=(V,\Sigma )$ consists of a vertex set $V$ and a collection $\Sigma$ of finite nonempty subsets containing every singleton and every nonempty subset of each member. A member with $n+1$ vertices is an $n$-simplex.

Its topological realization glues a standard geometric simplex for each member along common faces. A concrete model iswith finite support and the weak topology of a simplicial realization: a set is closed precisely when its intersection with every simplex is closed. This agrees with the quotient topology of the gluing construction. Specifying this topology matters for complexes with infinitely many vertices.

The open star in a simplicial complex of a vertex $v$ is the union of the relative interiors of the simplices containing $v$. In barycentric coordinates it isLet $L$ be the subcomplex consisting of all simplices not containing $v$. The complement of the star is exactly $|L|$: the positive-coordinate support of a point misses $v$ precisely when it is a simplex of $L$. By the allowed closed-subcomplex theorem, $|L|$ is closed in $|K|$. Hence the star is open.

This is the open-star convention required by the question. The closed star, which also includes the opposite faces, is a different subset and need not be open.

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.