how can I show that any finite CW-space can embedded into an euclidean space of some dimension? Any help or reference would be greatly appreciated.
-
1$\begingroup$ See epub.ub.uni-muenchen.de/4524/1/4524.pdf $\endgroup$– Vidit NandaOct 26, 2013 at 16:40
-
2$\begingroup$ This is proved in Hatcher's "Algebraic Topology" corollary A.10. Voted to close. $\endgroup$– Igor BelegradekOct 26, 2013 at 20:09
2 Answers
If your finite CW-complex is of topological dimension $n$, then it is an $n$-dimensional compact metric space, thus, by the The Menger-Nöbeling theorem (1932), it can be embedded in ${\mathbb R}^{2n+1}$. In this theorem $2n+1$ is the lowest possible dimension, since there exist $n$-dimensional simplicial complexes that cannot be embedded in ${\mathbb R}^{2n}$.
Well, any simplicial complex can be realized as a subset of the simplex in $\mathbb{R}^V$ (where $V$ is the number of vertices). But a CW complex can only be embedded up to homotopy, it seems (see the answer to your duplicate question on math.stackexchange)
-
1$\begingroup$ As long as $X$ is a separable metrizable space of finite dimension $n$, space $X$ can be topologically embedded in $\mathbb R^{2\cdot n+1}$. (Thus Wlodek's pointer is correct, and the author of the QUESTION doesn't even need to restrict the CW-complex to be finite--separability would be enough to assure embeddability). $\endgroup$ Apr 15, 2014 at 7:46