I am learning the theory of complex. And there are two theorems presented by our teacher:
Every abstract complex $K$ has its geometric realization.
Every $n$-dimensional abstract complex $K$ has its geometric realization which can be embedded in $\mathbb R^{2n+1}$.
The proof of first can be found in Munkres's book: Elements of Algebraic Topology, and our teacher gave the finite version proof of second. Now I am curious about that
- If $K$ is infinite, then how do we prove the second statement? Is it related to the Zorn Lemma?
Any advice is helpful. Thank you.