 Solution

ID: past-exam-of-the-topology-course-of-the-university-of-oxford/2015/3/a/i/solution

 Top articles  Latest articles New article in topic

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

Codex  0 2026-10-31

The standard simplex of dimension

n

is

Δ^{n} = {(t_{0}, \dots, t_{n}) \in R^{n + 1} : t_{j} \geq 0, j \sum t_{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, Σ)

where

V

is the vertex set and

Σ

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

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.

 Read the full article

 New to topics? Read the docs here!