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 $h^{‴}{U}^{‴}$ is the homotopy category for unbased spaces , and a natural transformation $\gamma :\mathrm{\Gamma }⟶Id$ that asssigns a $CW$-complex $\mathrm{\Gamma }X$ and a weak equivalence ${\gamma }_e:\mathrm{\Gamma }X⟶X<math>toanarbitraryspace$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 $\mathrm{\Gamma }f:\mathrm{\Gamma }X\to \mathrm{\Gamma }Y$ 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 $\mathrm{\Gamma }X$ of a sequence of cellular inclusions of $CW$-complexes $X_1,\mathrm{...},X_n$ , so that one obtains $X\equiv colim[X_i]$. As a consequence of J.H.C. Whitehead's Theorem, one also has that:

$\gamma \ast :[\mathrm{\Gamma }X,\mathrm{\Gamma }Y]⟶[\mathrm{\Gamma }X,Y]$ is an isomorphism.

Furthermore, the homotopy groups of the $CW$-complex $\mathrm{\Gamma }X$ are the colimits of the homotopy groups of $X_n$ and ${\gamma }_{n+1}:{\pi }_q(X_{n+1})⟼{\pi }_q(X)$ is a group epimorphism.

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