 Solution

ID: past-exam-of-the-topology-course-of-the-university-of-oxford/2014/3/a/i/solution

K = (V, Σ)

consists of a vertex set

V

and a collection

Σ

of finite nonempty subsets containing every singleton and every nonempty subset of each member. A member with

n + 1

n

Its topological realization glues a standard geometric simplex for each member along common faces. A concrete model is

∣ K ∣ = {(t_{v})_{v \in V} : t_{v} \geq 0, \sum_{v} t_{v} = 1, {v : t_{v} > 0} \in Σ},

with 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.

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