# 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 -complexes}):

"There is a functor

Failed to parse (syntax error): {\displaystyle \Gamma: "'hU''' \longrightarrow '''hU''' }where is the homotopy category for unbased spaces , and a natural transformation that asssigns a -complex and a weak equivalence 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 is unique up to homotopy equivalence.

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

The -complex specified in the approximation theorem for an arbitrary space is constructed as the colimit of a sequence of cellular inclusions of -complexes , so that one obtains . As a consequence of J.H.C. Whitehead's Theorem, one also has that:

is an isomorphism.

Furthermore, the homotopy groups of the -complex are the colimits of the homotopy groups of and is a group epimorphism.

\begin{thebibliography} {9}

</ref>^{[1]}</references>

- ↑
^{1.0}^{1.1}May, J.P. 1999,*A Concise Course in Algebraic Topology.*, The University of Chicago Press: Chicago