nLab
dimension of a cell complex

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Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

The dimension of a cell complex XX is the largest natural number nn \in \mathbb{N}, if such exists, for which there are non-trivial nn-cells in XX. If there is no such largest nn the dimension is also said to be infinite.

Definition

Specifically, if XX is a CW-complex

Xlim nX n X \;\simeq\; \underset{\longrightarrow_{\mathrlap{n}}}{\lim} X_n

where each X nX_n is a pushout of the form

iI nS n1 X n1 (po) iI nD n X n \array{ \underset{ i \in I_n }{\sqcup} S^{n-1} &\longrightarrow& X_{n-1} \\ \big\downarrow &(po)& \big\downarrow \\ \underset{ i \in I_n }{\sqcup} D^{n} &\longrightarrow& X_n }

the dimension of XX is the largest nn for which the indexing sets I nI_n are non-empty.

Created on February 26, 2019 at 04:36:46. See the history of this page for a list of all contributions to it.