Past exam of the topology course of the University of Oxford / 2015 / 3 / a / i / Solution

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.