This page is about a property of Čech nerves in homotopy theory. For the “nerve theorem” in category theory see at Segal conditions.
topology (point-set topology, point-free topology)
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closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
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The nerve theorem asserts that the homotopy type of a sufficiently nice topological space is encoded in the Cech nerve of a good cover.
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.
Let $X$ be a paracompact space and $\{U_i \to X\}$ a good open cover. Write $C(\{U_i\})$ for the Cech nerve of this cover
(a simplicial space) and write
for the simplicial set obtained by replacing in $C(\{U_i\})$ each direct summand space by the point. Let $|\tilde C(\{U_i\})|$ be the geometric realization.
This is homotopy equivalent to $X$.
This is usually attributed to (Borsuk1948). The proof relies on the existence of partitions of unity (see for instance the review Hatcher, prop. 4G.2).
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.)
Armand Borel and Jean-Pierre Serre, Corners and arithmetic groups, Comment. Math. Helv. 48 (1973), 436–491. (See Theorem 8.2.1.)
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)