approximation theorem for an arbitrary space

Theorem 0.1.

(Approximation theorem for an arbitrary topological spaceMathworldPlanetmath in terms of the colimitMathworldPlanetmath of a sequencePlanetmathPlanetmath of cellular inclusions of CW-complexes):

“There is a functorMathworldPlanetmath Γ:𝒉𝑼𝒉𝑼 where hU is the homotopy category for unbased spaces , and a natural transformation γ:ΓId that asssigns a CW-complex ΓX and a weak equivalenceMathworldPlanetmath γe:ΓXX to an arbitrary space X, such that the following diagram commutes:

ΓXΓfΓY γ(X)γ(Y)X@ >fY

and Γf:ΓXΓY is unique up to homotopy equivalenceMathworldPlanetmathPlanetmath.”

(viz. p. 75 in ref. [1]).

Remark 0.1.

The CW-complex specified in the approximation theorem for an arbitrary space ( is constructed as the colimit ΓX of a sequence of cellular inclusions of CW-complexes X1,,Xn , so that one obtains Xcolim[Xi]. As a consequence of J.H.C. Whitehead’s Theorem, one also has that:

γ*:[ΓX,ΓY][ΓX,Y] is an isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

Furthermore, the homotopy groupsMathworldPlanetmath of the CW-complex ΓX are the colimits of the homotopy groups of Xn and γn+1:πq(Xn+1)πq(X) is a group epimorphism.


  • 1 May, J.P. 1999, A Concise Course in Algebraic Topology., The University of Chicago Press: Chicago
Title approximation theorem for an arbitrary space
Canonical name ApproximationTheoremForAnArbitrarySpace
Date of creation 2013-03-22 18:14:40
Last modified on 2013-03-22 18:14:40
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 43
Author bci1 (20947)
Entry type Theorem
Classification msc 81T25
Classification msc 81T05
Classification msc 81T10
Classification msc 55U15
Classification msc 57Q05
Classification msc 57Q55
Classification msc 55U05
Classification msc 55U10
Synonym approximation theorems for topological spaces
Related topic TheoremOnCWComplexApproximationOfQuantumStateSpacesInQAT
Related topic CWComplex
Related topic SpinNetworksAndSpinFoams
Related topic HomotopyCategory
Related topic WeakHomotopyEquivalence
Related topic GroupHomomorphism
Related topic ApproximationTheoremAppliedToWhitneyCrMNSpaces
Defines unique colimit of a sequence of cellular inclusions of CW-complexes