Past exam of the topology course of the University of Oxford / 2017 / 3 / a / i / 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 ${\lambda }_v\ge 0$ 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]