# 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 ${\displaystyle CW}$-complexes}):

"There is a functor $\displaystyle \Gamma: "'hU''' \longrightarrow '''hU'''$ where ${\displaystyle '''hU'''}$ is the homotopy category for unbased spaces , and a natural transformation ${\displaystyle \gamma :\Gamma \longrightarrow Id}$ that asssigns a ${\displaystyle CW}$-complex ${\displaystyle \Gamma X}$ and a weak equivalence ${\displaystyle \gamma _{e}:\Gamma X\longrightarrow X$toanarbitraryspace}$X$, such that the following diagram commutes:

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

(viz. p. 75 in ref. [1]). \end{theorem}

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

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

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

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</ref>[1]</references>

1. May, J.P. 1999, A Concise Course in Algebraic Topology. , The University of Chicago Press: Chicago