nLab
connected filtered space

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

Connected filtered spaces

Definition

A filtered space X *X_* is called a connected filtered space if it satisfies:

  1. (ϕ) 0(\phi)_0: The function π 0X 0π 0X r\pi_0X_0 \to \pi_0X_r induced by inclusion is surjective for all r>0r \gt 0; and,

  2. for all i1i \geq 1, (ϕ i):π i(X r,X i,v)=0(\phi_i): \pi_i(X_r,X_i,v)=0 for all r>ir \gt i and vX 0 v \in X_0.

Another equivalent form is:

  1. (ϕ 0)(\phi_0'): The function π 0X sπ 0X r\pi_0X_s \to \pi_0X_r induced by inclusion is surjective for all 0=s<r0=s \lt r and bijective for all 1sr1 \leq s \leq r; and,

  2. for all i1i \geq 1, (ϕ i):π j(X r,X i,v)=0(\phi_i'): \, \pi_j(X_r,X_i,v)=0 for all vX 0v \in X_0 and all j,r j,r such that 1ji<r 1 \leq j \leq i \lt r.

Examples

Examples of connected filtered spaces are:

  1. The skeletal filtration of a CW-complex.

  2. The word length filtration of the James construction for a space with base point such that {x}X\{x\} \to X is a closed cofibration.

  3. The filtration (BC) *(B C)_* of the classifying space of a crossed complex, filtered using skeleta of CC.

This condition occurs in the higher homotopy van Kampen theorem for crossed complexes.

Revised on July 22, 2014 12:16:11 by Toby Bartels (98.23.133.202)