This can be seen as a special case of some aspects of étale homotopy as the étale homotopy type of nice spaces coincides with the homotopy type of its Cech nerve.
This is homotopy equivalent to .
This statement implies that in the cohesive (∞,1)-topos ETop∞Grpd the intrinsic fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos coincides with the ordinary fundamental ∞-groupoid functor of paracompact topological spaces. See Euclidean-topological ∞-groupoid : Geometric homotopy for details.
Some earlier references on the nerve theorem include
Jean Leray, L’anneau spectral et l’anneau filtré d’homologie d’un espace localement compact et d’une application continue, J. Math. Pures Appl. (9) 29 (1950), 1–139.
André Weil, Sur les theoremes de de Rham, Comment. Math. Helv. 26 (1952), 119–145. (See §6.)
Michael C. McCord, Homotopy type comparison of a space with complexes associated with its open covers, Proc. Amer. Math. Soc. 18 (1967), 705–708.
Graeme Segal, Classifying spaces and spectral sequences, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 105–112. (See §4.)
A version for hypercovers is discussed in
A review appears as corollary 4G.3 in the textbook
Some slightly stronger statements are discussed in
Anders Björner, Nerves, fibers and homotopy groups , Journal of combinatorial theory, series A, 102 (2003), 88-93
Andrzej Nagórko, Carrier and nerve theorems in the extension theory Proc. Amer. Math. Soc. 135 (2007), 551-558. (web)