PlanetPhysics/Approximation Theorem for an Arbitrary Space

\begin{theorem}(Approximation \htmladdnormallink{theorem {http://planetphysics.us/encyclopedia/Formula.html} for an arbitrary topological space in terms of the colimit of a sequence of cellular inclusions of $CW$-complexes}):

"There is a functor Failed to parse (syntax error): {\displaystyle \Gamma: "'hU''' \longrightarrow '''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 $\gamma _{e}:\Gamma X\longrightarrow X<math>toanarbitraryspace$X</math>, such that the following diagram commutes:

Failed to parse (unknown function "\begin{CD}"): {\displaystyle \begin{CD} \Gamma X @> \Gamma f >> \Gamma Y \\ @V <math>~~~~~~~} \gamma (X) VV @VV \gamma (Y) V \\ X @ > f >> Y \end{CD} </math> and $\Gamma f:\Gamma X\rightarrow \Gamma Y$ is unique up to homotopy equivalence.

(viz. p. 75 in ref. ). \end{theorem}

The $CW$-complex specified in the approximation theorem for an arbitrary space 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.

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