CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
A cover $\{U_i \to X\}$ of a topological space $X$ is called a good cover – or good open cover if it is
an open cover;
such that all the $U_i$ and all their nonempty finite intersections are contractible topological spaces.
For $X$ a topological manifold one often needs that the non-empty finite intersections are homeomorphic to an open ball and for smooth manifolds often one needs that the finite non-empty intersections are diffeomorphic to an open ball.
In the literature this is traditionally glossed over, but it is in fact a subtle point, see the discussion at open ball and see below at Existence on paracompact manifolds.
Therefore it makes sense to explicitly say:
Given a smooth manifold $X$ a differentiably good open cover is a good open cover one all whose finite non-empty intersections are in fact diffeomorphic to an open ball, hence to a Cartesian space.
Every paracompact manifold admits a good open cover, def. 1.
Every paracompact manifold admits a Riemannian metric, and for any point in a Riemannian manifold there is a geodesically convex neighborhood (any two points in the neighborhood are connected by a unique geodesic in the neighborhood, one whose length is the distance between the points; see for example the remark after lemma 10.3 (Milnor) page 59). It is immediate that a nonempty intersection of finitely many such geodesically convex neighborhoods is also geodesically convex, hence contractible.
It is apparently a folk theorem that every geodesically convex neighbourhood in a Riemannian manifold is diffeomorphic to a Cartesian space (for instance this is asserted on page 42 of (BottTu). This implies the following strengthening of the above result, which appears stated as theorem 5.1 in BottTu. But a complete proof in the literature is hard to find (see also the discussion of the references at ball). We give a complete proof below.
Every paracompact manifold of dimension $d$ admits a differentiably good open cover, def. 2, hence an open cover such that every non-empty finite intersection is diffeomorphic to the Cartesian space $\mathbb{R}^d$.
By (Greene) every paracompact manifold admits a Riemannian metric with positive convexity radius $r_{\mathrm{conv}} \in \mathbb{R}$. Choose such a metric and choose an open cover consisting for each point $p\in X$ of the geodesically convex open subset $U_p := B_p(r_{conv})$ given by the geodesic $r_{conv}$-ball at $p$. Since the injectivity radius of any metric is at least $2r_{\mathrm{conv}}$ it follows from the minimality of the geodesics in a geodesically convex region that inside every finite nonempty intersection $U_{p_1} \cap \cdots \cap U_{p_n}$ the geodesic flow around any point $u$ is of radius less than or equal the injectivity radius and is therefore a diffeomorphism onto its image.
Moreover, the preimage of the intersection region under the geometric flow is a star-shaped region in the tangent space $T_u X$: because the intersection of geodesically convex regions is itself geodesically convex, so that for any $v \in T_u X$ with $\exp(v) \in U_{p_1} \cap \cdots \cap U_{p_n}$ the whole geodesic segment $t \mapsto \exp(t v)$ for $t \in [0,1]$ is also in the region.
So we have that every finite non-empty intersection of the $U_p$ is diffeomorphic to a star-shaped region in a vector space. By the results cited at ball (e.g. theorem 237 of (Ferus)) this star-shaped region is diffeomorphic to an $\mathbb{R}^n$.
The category $ParaMfd$ of paracompact manifolds admits a coverage whose covering families are good open covers.
The same holds true for subcategories such as
Diff – paracompact smooth manifolds;
It is sufficient to check this in $ParaMfd$. We need to check that for $\{U_i \to U\}$ a good open cover and $f : V \to U$ any morphism, we get commuting squares
such that the $\{V_i \to V\}$ form a good open cover of $V$.
Now, while $ParaMfd$ does not have all pullbacks, the pullback of an open cover does exist, and since $f$ is necessarily a continuous function this is an open cover $\{f^* U_i \to V\}$. The $f^* U_i$ need not be contractible, but being open subsets of a paracompact manifold, they are themselves paracompact manifolds and hence admit themselves good open covers $\{W_{i,j} \to f^* U_i\}$.
Then the family of composites $\{W_{i,j} \to f^* U_i \to V\}$ is clearly a good open cover of $V$.
Every CW complex admits a good open cover.
It suffices to prove that if $X$ admits a good open cover and $\phi: S^n \to X$ is an attaching map, then the pushout
also admits a good open cover. Let $\{U_\alpha\}$ be a good open cover of $X$ closed under nonempty finite intersections, and choose a contracting homotopy $h_\alpha: I \times U_\alpha \to U_\alpha$ such that $h_\alpha(0, -) = id$ and $h_\alpha(1, -)$ is constant. For any subset $S \subseteq D^{n+1}$, let $Hull(S)$ denote the convex hull of $S$. Then, if $V$ is relatively open in the boundary $S^n$, $Hull(V)$ is open in $D^{n+1}$. It follows that the image in $Y$ of
is open in $Y$, and it is contractible: define a contracting homotopy
by
These sets $V_\alpha$ together with $int(D^{n+1})$ form a good open cover of $Y$.
The following nPOV perspective on good open covers gives a useful general “explanation” for their relevance, which also explains the role of good covers in Cech cohomology generally and abelian sheaf cohomology in particular.
Let $sPSh(CartSp)_{proj}$ be the category of simplicial presheaves on the category CartSp equipped with the projective model structure on simplicial presheaves.
Let $X$ be a smooth manifold, regarded as a 0-truncated object of $sPSh(C)$.
Let $\{U_i \to X\}$ be a good open cover by open balls in the strong sense: such that every finite non-empty intersection is diffeomorphic to an $\mathbb{R}^d$.
Then: the Cech nerve $C(\{U\}) \in sPSh(C)$ is a cofibrant resolution of $X$ in the local model structure on simplicial presheaves.
By assumption we have that $C(U)$ is degreewise a coproduct of representables. It is also evidently a split hypercover.
This implies the statement by the characterization of cofibrant objects in the projective structure.
This has a useful application in the nerve theorem.
Notice that the descent condition for simplicial presheaves on CartSp at (good) covers is very weak, since all Cartesian spaces are topologically contractible, so it is easy to find the fibrant objects $A \in sPSh(C)_{proj, loc}$ in the topological localization of $sPSh(C)_{proj}$ for the canonical coverage of CartSp. The above observation says that for computing the $A$-valued cohomology of a diffeological space $X$, it is sufficient to evaluate $A$ on (the Cech nerve of) a good cover of $X$.
We can turn this around and speak for any site $C$ of a covering family $\{U_i \to X\}$ as being good if the corresponding Cech nerve is degreewise a coproduct of representables. In the projective model structure on simplicial presheaves on $C$ such good covers will enjoy the central properties of good covers of topological spaces.