connected filtered space



topology (point-set topology)

see also algebraic topology, functional analysis and homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Basic homotopy theory

Connected filtered spaces


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 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 (