# approximation theorem for an arbitrary space

###### Theorem 0.1.

(Approximation theorem for an arbitrary topological space in terms of the colimit of a sequence of cellular inclusions of $CW$-complexes):

“There is a functor $\Gamma:\textbf{hU}\longrightarrow\textbf{hU}$ where hU is the homotopy category for unbased spaces , and a natural transformation $\gamma:\Gamma\longrightarrow Id$ that asssigns a $CW$-complex $\Gamma X$ and a weak equivalence $\gamma_{e}:\Gamma X\longrightarrow X$ to an arbitrary space $X$, such that the following diagram commutes:

 $\begin{CD}\Gamma X@>{\Gamma f}>{}>\Gamma Y\\ @V{~{}~{}~{}~{}~{}~{}~{}\gamma(X)}V{}V@V{}V{\gamma(Y)}V\\ X@ >f>>Y\end{CD}$

and $\Gamma f:\Gamma X\rightarrow\Gamma Y$ is unique up to homotopy equivalence.”

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

###### Remark 0.1.

The $CW$-complex specified in the approximation theorem for an arbitrary space (http://planetmath.org/ApproximationTheoremForAnArbitrarySpace) is constructed as the colimit $\Gamma X$ of a sequence of cellular inclusions of $CW$-complexes $X_{1},...,X_{n}$ , so that one obtains $X\equiv colim[X_{i}]$. As a consequence of J.H.C. Whitehead’s Theorem, one also has that:

$\gamma*:[\Gamma X,\Gamma Y]\longrightarrow[\Gamma X,Y]$ is an isomorphism.

Furthermore, the homotopy groups of the $CW$-complex $\Gamma X$ are the colimits of the homotopy groups of $X_{n}$ and $\gamma_{n+1}:\pi_{q}(X_{n+1})\longmapsto\pi_{q}(X)$ is a group epimorphism.

## References

• 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