= Solution
An <abstract simplicial complex> is a pair $K=(V,\Sigma)$, where $V$ is a <set> of vertices and $\Sigma$ is a family of finite nonempty <subsets> of $V$ satisfying two conditions: every singleton $\{v\}$ is in $\Sigma$, and every nonempty subset of a member of $\Sigma$ is again in $\Sigma$. Its members are the <simplexes>; a face with $n+1$ vertices has dimension $n$.
To form its <topological realization>, take a disjoint copy of the standard geometric simplex $\Delta^{|\sigma|-1}$ for every $\sigma\in\Sigma$, label its vertices by $\sigma$, and identify shared labelled faces using affine face inclusions. Give the resulting space $|K|$ the <quotient topology> of this disjoint union. Equivalently, its points have finite-support <barycentric coordinates> $\lambda_v\geq0$ with $\sum_v\lambda_v=1$ and positive support belonging to $\Sigma$.
The topology is part of the definition: $U\subseteq|K|$ is open exactly when $U\cap|\sigma|$ is open in each closed geometric simplex. This is the <weak topology of a simplicial realization>, used below for the infinite complex. For finite complexes it coincides with the usual Euclidean realization topology.<Topology and Groups by Marc Lackenby>[Definitions I.17–I.18, printed pp. 8–9: abstract complexes and face-gluing realization]{ref}
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