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 $$\Gamma: "'hU \longrightarrow hU''' $$ 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 $$\gamma _e:\Gamma X \longrightarrow XX$$, such that the following diagram commutes:

$$ \begin{CD} \Gamma X @> \Gamma f >> \Gamma Y \\ @V ~\gamma (X) VV @VV \gamma (Y) V \\ X @ > f >> Y \end{CD} $$ and $$\Gamma f: \Gamma X\rightarrow \Gamma Y$$ is unique up to homotopy equivalence. (viz. p. 75 in ref. ). \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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