I see a theorem without proof on Gelfand & Manin:
Suppose $\mathfrak U=\{U_\alpha\}_\alpha$ is a locally finite open covering of the topological space $X$ such that each finite intersection $U_{\alpha_1}\cap\dotsc\cap U_{\alpha_n}$ is contractible. Then $X$ is homotopically equivalent to the Alexandrov-Čech nerve $N(\mathfrak U)$.
I need a reference of a sketch of a proof for the preceding theorem. In fact, the following theorem (Theorem 13.4) of Bott & Tu is a corollary of this:
Suppose the topological space $X$ has a good cover $\mathfrak U$. Then the fundamental group of $X$ is isomorphic to the fundamental group $\pi_1(N(\mathfrak U))$ of the nerve of the good cover.
Maybe the idea of the proof also applies to the stronger theorem.
Any idea? Thanks!
There's a proof of this fact (in the category of CW-complexes, at least) in section 4.G of Hatcher's "Algebraic Topology." The basic idea is to construct an explicit map $X\to N(\mathfrak{U})$, then use Whitehead's theorem to show that it's a homotopy equivalence. The proof uses paracompactness, but it's only used to construct a partition of unity; the local finiteness condition gives you that for free.