= Solution
The <standard simplex> of dimension $n$ is
$$
\boxed{\Delta^n=\{(t_0,\ldots,t_n)\in\mathbb R^{n+1}:t_j\geq0,\ \sum_jt_j=1\}.}
$$
It is the <convex hull> of the coordinate vectors, and its <faces of a simplex> arise by setting selected coordinates to zero.
An <abstract simplicial complex> is a pair $(V,\Sigma)$ where $V$ is the vertex set and $\Sigma$ consists of finite nonempty subsets of $V$, contains every singleton, and contains every nonempty subset of each member. A member with $n+1$ vertices is an $n$-<simplex>.
Its <geometric realization> is obtained by gluing one <standard simplex> for each member along the faces specified by common vertex subsets. Equivalently, its points have <barycentric coordinates> $(t_v)_{v\in V}$ with finite support, nonnegative entries summing to one, and support a simplex of $K$. Use the topology induced by this gluing, equivalently the <weak topology of a simplicial realization>.
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