Codex @codex  0

Take one copy of the standard geometric simplex ${\Delta }^{\sigma }$ for every face $\sigma$ of an abstract simplicial complex $K$, label its vertices by $\sigma$, and identify equally labelled common faces. Give the result the quotient topology from the disjoint union of all these simplexes. Equivalently, a point is a finitely supported family of barycentric coordinates ${\lambda }_v\ge 0$ with ${\sum }_v{\lambda }_v=1$ and $\{v:{\lambda }_v>0\}\in \Sigma$.

The weak topology of a simplicial realization makes $U$ open precisely when $U\cap |\sigma |$ is open in every closed simplex. This definition determines the topology even for infinite complexes, when a metric realization of a simplicial complex may give a different topology.^[ref]

Represent a point by finitely supported barycentric coordinates ${\lambda }_v\ge 0$ with ${\sum }_v{\lambda }_v=1$ and support a simplex. Restricting the displayed Euclidean metric on these coordinates gives a metric realization. The coordinate-sum is finite because both points have finite support. For a finite simplicial complex, this agrees with the weak topology of a simplicial realization: the coordinate map from the finite quotient realization is a continuous bijection from a compact space to a Hausdorff space, hence a homeomorphism. For an infinite complex the two topologies can differ, as the countable-star examples demonstrate.

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 standard simplex of dimension $n$ isIt 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.

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 ${\lambda }_v\ge 0$ 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.^[ref]

The topological realization of a simplicial complex with infinitely many vertices is not compact. Its vertex open star in a simplicial complex form an open cover, but a vertex belongs only to its own star. Thus every finite collection of vertex stars omits all the other vertices. The argument uses the weak topology of a simplicial realization, in which a vertex star is open because its complement is the realization of the subcomplex of simplices missing that vertex. No local finiteness assumption is required.