= Solution
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 is
$$
|K|=\left\{(t_v)_{v\in V}:t_v\geq0,\ \sum_vt_v=1,\ \{v:t_v>0\}\in\Sigma\right\},
$$
with 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.
Back to article page
