81T25-ApproximationTheoremForAnArbitrarySpace.tex

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\pmtitle{approximation theorem for an arbitrary space}
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\pmdefines{unique colimit of a sequence of cellular inclusions of  $CW$-complexes}

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\begin{document}
\begin{theorem}({\em Approximation theorem for an arbitrary topological space in terms of the colimit of a sequence of cellular inclusions of $CW$-complexes}): 

\begin{quote}
``There is a functor $\Gamma: \textbf{hU} \longrightarrow \textbf{hU}$ where 
$\textbf{hU}$ 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 $\gamma _e:\Gamma X \longrightarrow 
X$ to an arbitrary space $X$, 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.''
\end{quote}
(viz. p. 75 in ref. \cite{MJP1999}).
\end{theorem}

\begin{remark}
  The $CW$-complex specified in the 
\PMlinkname{approximation theorem for an arbitrary space}{ApproximationTheoremForAnArbitrarySpace} 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.
\end{remark}

\begin{thebibliography} {9}

\bibitem{MJP1999}
May, J.P. 1999, \emph{A Concise Course in Algebraic Topology.}, The University of Chicago Press: Chicago

\end{thebibliography}
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\end{document}