## You are here

Homeapproximation theorem for an arbitrary space

## Primary tabs

# approximation theorem for an arbitrary space

###### Theorem 0.1.

(Approximation theorem for an arbitrary topological space in terms of the colimit of a sequence of cellular inclusions of $CW$-complexes):

“There is a functor $\Gamma:\textbf{hU}\longrightarrow\textbf{hU}$ 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 $\gamma_{e}:\Gamma X\longrightarrow X$ to an arbitrary space $X$, such that the following diagram commutes:

$\begin{matrix}\Gamma X&\cd@stack{\rightarrowfill@}{\Gamma f}{}&\Gamma Y\\ {$~{}~{}~{}~{}~{}~{}~{}$\gamma(X)}{\Big\downarrow}&&{}{\Big\downarrow}{\gamma(% Y)}&&\\ X@ >f>>Y\end{matrix}$ and $\Gamma f:\Gamma X\rightarrow\Gamma Y$ is unique up to homotopy equivalence.”

(viz. p. 75 in ref. [1]).

###### Remark 0.1.

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.

# References

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

## Mathematics Subject Classification

81T25*no label found*81T05

*no label found*81T10

*no label found*55U15

*no label found*57Q05

*no label found*57Q55

*no label found*55U05

*no label found*55U10

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections