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subsets are closed in a closed subspace precisely if they are closed in the ambient space
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open subspaces of compact Hausdorff spaces are locally compact
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paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
Theorems
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 inhabited finite intersections are contractible topological spaces.
For $X$ a topological manifold one often requires that the inhabited finite intersections are homeomorphic to an open ball. Similarly, for $X$ a smooth manifold one often requires that the finite inhabited intersections are diffeomorphic to an open ball.
In the literature this is traditionally glossed over, but this is in fact a subtle point, see the discussion at open ball and see below at Existence on paracompact smooth manifolds.
Due to this subtly, it is instructive to make explicit the following definition:
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 smooth manifold admits a good open cover, def. 1, in fact a differentiable good open cover, def. 2
Every paracompact smooth 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 (Milnor, lemma 10.3 on page 59), or (do Carmo, Proposition 4.2)). A nonempty intersection of finitely many such geodesically convex neighborhoods is also geodesically convex. The inverse of the exponential map based at any interior point of a geodesically convex open subset gives a diffeomorphism from this subset to a star-shaped open subset of $\mathbf{R}^n$. Indeed, the Gauss lemma shows that the tangent map of the exponential map is invertible. By definition of geodesic convexity the exponential map is injective, hence a diffeomorphism. By this theorem, star-shaped open subsets of $\mathbf{R}^n$ are diffeomorphic to $\mathbf{R}^n$, which completes the proof.
It is apparently a folk theorem that every geodesically convex open neighbourhood in a Riemannian manifold is diffeomorphic to a Cartesian space. For instance, this is asserted in the proof of Theorem 5.1 of (BottTu), which claims the existence of differentiable good open covers. But a complete proof in the literature is hard to find. See this remark at the discussion of the references at open ball.
Every smooth 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 smooth 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 $ParaSmMfd$ of paracompact smooth manifolds admits a coverage whose covering families are good open covers.
The same holds true for full subcategories such as
It is sufficient to check this in $ParaSmMfd$. 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 $ParaSmMfd$ 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 finite CW complex admits a good open cover.
Hopefully someone can find a clear reference to a proof. The assertion for finite CW complexes is found for example here (Topology of Tiling Spaces by Sadun, p. 37). It is not immediately clear from the remarks there what obstructions would exist to generalizing the assertion to all CW complexes.
As indicated at CW complex, every CW complex is homotopy equivalent to a simplicial complex, and simplicial complexes certainly admit good covers by taking open stars.
For a (paracompact) topological manifold the construction via Riemannian metrics or similar smooth constructions in general does not work.
In (Osborne-Stern 69) the following discussion for sufficient conditions getting “close” to good open covers is discussed:
Let $X$ be a k-connected topological manifold of dimension $n$ (without boundary), and define
For $p \in \mathbb{N}$ such that $p(q+1) \gt n$ then $X$ admits a cover by $p$ open balls and such that all nonempty intersections of the covering cells are (q−1)-connected.
A cover $\{U_i\to X\}_{i\in I}$ refines another cover $\{V_j\to X\}_{j\in J}$ if each map $V_j\to X$ is some $U_i\to X$.
Each differentially good cover has a unique smallest refinement to a differentially good cover that is closed under intersection.
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.
Manfredo do Carmo, Riemannian geometry (trans. Francis Flaherty), Birkhäuser (1992)
John Milnor, Morse theory , Princeton University Press (1963)
R. Greene, Complete metrics of bounded curvature on noncompact manifolds Archiv der Mathematik Volume 31, Number 1 (1978)
Dirk Ferus, Analysis III (pdf)
Raoul Bott, Loring Tu, Differential forms in algebraic topology, Graduate texts in mathematics vol. 82 (1982) (pdf)
RP Osborne and JL Stern. Covering Manifolds with Cells, Pacific Journal of Mathematics, Vol 30, No. 1, 1969.
MathOverflow, Proving the existence of good covers