= Solution
An abstract <simplicial complex> consists of a <set> $V$ of <vertices of a simplicial complex> and a family $K$ of finite nonempty <subsets> of $V$, called abstract <simplexes>, such that every <singleton> $\{v\}$ belongs to $K$ and every nonempty <subset> of a member of $K$ also belongs to $K$. One may include the <empty set> as an additional face by convention. A member with $k+1$ vertices is a $k$-<simplex>; its nonempty <subsets> are its <faces of a simplex>.
The <geometric realization> has the following explicit description using a <barycentric coordinate system>:
$$
|K|=\left\{t:V\to[0,1]:\operatorname{supp}(t)\text{ is finite},\ \sum_{v\in V}t(v)=1,\ \operatorname{supp}(t)\in K\right\}.
$$
Here the <support> is $\operatorname{supp}(t)=\{v:t(v)>0\}$. Each $\rho\in K$ gives a standard geometric <simplex>
$$
|\rho|=\{t\in|K|:t(v)=0\text{ for }v\notin\rho\},
$$
with its usual <Euclidean topology>. Common <faces of a simplex> are identified using the same <barycentric coordinate system>.
Precisely, the <topology> on $|K|$ is defined by declaring $A\subseteq|K|$ <closed> if and only if $A\cap|\rho|$ is <closed> in $|\rho|$ for every $\rho\in K$. Equivalently, use the <quotient topology> on the <disjoint union> of all the standard geometric <simplexes>, identifying identical points of common <faces of a simplex>. For a finite <simplicial complex>, this agrees with the <subspace topology> in $\mathbb R^V$. For an infinite <simplicial complex>, the stated rule specifies the realization <topology> without presuming a finite-dimensional ambient space.
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