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\section*{good open cover}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

\hypertarget{topos_theory}{}\paragraph*{{Topos theory}}\label{topos_theory}

[[!include topos theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{ExistenceOnParacompactManifolds}{Existence on paracompact smooth manifolds}\dotfill \pageref*{ExistenceOnParacompactManifolds} \linebreak
\noindent\hyperlink{coverages_of_good_open_covers}{Coverages of good open covers}\dotfill \pageref*{coverages_of_good_open_covers} \linebreak
\noindent\hyperlink{existence_on_cw_complexes}{Existence on CW complexes}\dotfill \pageref*{existence_on_cw_complexes} \linebreak
\noindent\hyperlink{nonexistence_for_topological_manifolds}{(Non-)Existence for topological manifolds}\dotfill \pageref*{nonexistence_for_topological_manifolds} \linebreak
\noindent\hyperlink{refining_covers}{Refining covers}\dotfill \pageref*{refining_covers} \linebreak
\noindent\hyperlink{pov}{$n$POV}\dotfill \pageref*{pov} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{GoodOpenCover}\hypertarget{GoodOpenCover}{}
A [[cover]] $\{U_i \to X\}$ of a [[topological space]] $X$ is called a \textbf{good cover} -- or \textbf{good open cover} if it is

\begin{enumerate}%
\item an [[open cover]];


\item such that all the $U_i$ and all their inhabited finite intersections are [[contractible topological spaces]].



\end{enumerate}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
For $X$ a [[topological manifold]] one often requires that the inhabited finite intersections are [[homeomorphism|homeomorphic]] to an [[open ball]]. Similarly, for $X$ a [[smooth manifold]] one often requires that the finite inhabited intersections are [[diffeomorphism|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 \emph{\hyperlink{ExistenceOnParacompactManifolds}{Existence on paracompact smooth manifolds}}.

\end{remark}
Due to this subtlety, it is instructive to make explicit the following definition:

\begin{defn}
\label{DifferentiablyGoodOpenCover}\hypertarget{DifferentiablyGoodOpenCover}{}
Given a [[smooth manifold]] $X$ a \emph{differentiably good open cover} is a good open cover one all whose finite non-empty intersections are in fact [[diffeomorphism|diffeomorphic]] to an [[open ball]], hence to a [[Cartesian space]].

\end{defn}
\hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties}

\hypertarget{ExistenceOnParacompactManifolds}{}\subsubsection*{{Existence on paracompact smooth manifolds}}\label{ExistenceOnParacompactManifolds}

\begin{prop}
\label{}\hypertarget{}{}
Every [[paracompact manifold|paracompact]] [[smooth manifold]] admits a good open cover, def.~\ref{GoodOpenCover}, in fact a differentiable good open cover, def.~\ref{DifferentiablyGoodOpenCover}

\end{prop}
\begin{proof}
Every [[paracompact manifold|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 (\hyperlink{Milnor}{Milnor, lemma 10.3 on page 59}), or (\hyperlink{doCarmo}{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 \href{open+ball#StarShapedOpenDiffeomorphicToOpenBall}{this theorem}, star-shaped open subsets of $\mathbf{R}^n$ are diffeomorphic to $\mathbf{R}^n$, which completes the proof.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
It is apparently a [[folk theorem]] that every [[geodesic convexity|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 (\hyperlink{BottTu}{BottTu}), which claims the existence of differentiable good open covers. But a complete proof in the literature is hard to find. See \hyperlink{LiteratureOnStarShapedOpenDiffeoToOpenBall}{this remark} at the discussion of the references at \emph{[[open ball]]}.

\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
Every [[smooth manifold|smooth]] [[paracompact manifold]] of [[dimension]] $d$ admits a differentiably good open cover, def. \ref{DifferentiablyGoodOpenCover}, hence an open cover such that every non-empty finite intersection is [[diffeomorphic]] to the [[Cartesian space]] $\mathbb{R}^d$.

\end{prop}
\begin{proof}
By (\hyperlink{Greene}{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 (\hyperlink{Ferus}{Ferus})) this star-shaped region is diffeomorphic to an $\mathbb{R}^n$.

\end{proof}
\hypertarget{coverages_of_good_open_covers}{}\subsubsection*{{Coverages of good open covers}}\label{coverages_of_good_open_covers}

\begin{prop}
\label{GoodOpenCoversFormACoverageOnParacompactSmooothManifolds}\hypertarget{GoodOpenCoversFormACoverageOnParacompactSmooothManifolds}{}
The [[category]] $ParaSmMfd$ of [[paracompact topological space|paracompact]] [[smooth manifolds]] admits a [[coverage]] whose covering families are good open covers.

The same holds true for [[full subcategories]] such as

\begin{itemize}%
\item [[CartSp]] -- [[Cartesian spaces]].

\end{itemize}
\end{prop}
\begin{proof}
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

\begin{displaymath}
\itexarray{  
     V_j &\to& U_{i(j)}
     \\
     \downarrow && \downarrow
     \\
     V &\stackrel{f}{\to}& U
  }
\end{displaymath}
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 $V$.

\end{proof}
\hypertarget{existence_on_cw_complexes}{}\subsubsection*{{Existence on CW complexes}}\label{existence_on_cw_complexes}

\begin{theorem}
\label{}\hypertarget{}{}
Every finite [[CW complex]] admits a good open cover.

\end{theorem}
Hopefully someone can find a clear reference to a proof. The assertion for finite CW complexes is found for example \href{https://books.google.com/books?id=dL8FCAAAQBAJ&pg=PA37&lpg=PA37&dq=%22good+cover%22+%22CW+complex%22&source=bl&ots=GrZW-ZGSpU&sig=eLwTeBNnBYJuEYBJ-6JCPTPvw1g&hl=en&sa=X&ved=0CDgQ6AEwBGoVChMIlJCJzIOGyAIVCmg-Ch2FQwlj#v=onepage&q=%22good%20cover%22%20%22CW%20complex%22&f=false}{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 \emph{homotopy equivalent} to a simplicial complex, and simplicial complexes certainly admit good covers by taking open stars.

\hypertarget{nonexistence_for_topological_manifolds}{}\subsubsection*{{(Non-)Existence for topological manifolds}}\label{nonexistence_for_topological_manifolds}

For a (paracompact) [[topological manifold]] the construction via [[Riemannian metrics]] or similar smooth constructions in general does not work.

In (\hyperlink{OsborneStern69}{Osborne-Stern 69}) the following discussion for sufficient conditions getting ``close'' to good open covers is discussed:

Let $X$ be a [[n-connected space|k-connected]] [[topological manifold]] of [[dimension]] $n$ (without [[boundary]]), and define

\begin{displaymath}
q \coloneqq
  min(k,n-3)
  \,.
\end{displaymath}
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 [[n-connected space|(q−1)-connected]].

\hypertarget{refining_covers}{}\subsubsection*{{Refining covers}}\label{refining_covers}

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.

\hypertarget{pov}{}\subsection*{{$n$POV}}\label{pov}

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.

\begin{prop}
\label{}\hypertarget{}{}
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 ball]]s 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]].

\end{prop}
\begin{proof}
By assumption we have that $C(U)$ is degreewise a [[coproduct]] of [[representable functor|representables]]. It is also evidently a [[split hypercover]].

This implies the statement by the \href{http://ncatlab.org/nlab/show/model+structure+on+simplicial+presheaves#CofibrantObjects}{characterization of cofibrant objects in the projective structure}.

\end{proof}
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 space]]s 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 \emph{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.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item \href{Stein+manifold#GoodCoversBySteinManifolds}{good covers by Stein manifolds} - note that this is a different concept, with vanishing Dolbeault cohomology replacing contractibility.

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

A fairly detailed proof is presented in Section 5.3 and Appendix C of

\begin{itemize}%
\item [[Victor Guillemin]], [[Peter Haine]], \emph{Differential Forms}, World Scientific (2019).

\end{itemize}
A similar proof appears in Lemma IV.6.9 of

\begin{itemize}%
\item [[Jean-Pierre Demailly]], \emph{Complex Analytic and Differential Geometry}

\end{itemize}
Other references include

\begin{itemize}%
\item [[Manfredo do Carmo]], \emph{Riemannian geometry} (trans. Francis Flaherty), Birkh\"a{}user (1992)


\item [[John Milnor]], \emph{Morse theory} , Princeton University Press (1963)


\item R. Greene, \emph{Complete metrics of bounded curvature on noncompact manifolds} Archiv der Mathematik Volume 31, Number 1 (1978)


\item [[Dirk Ferus]], \emph{Analysis III} (\href{http://www.math.tu-berlin.de/~ferus/ANA/Ana3.pdf}{pdf})


\item [[Raoul Bott]], [[Loring Tu]], \emph{Differential forms in algebraic topology}, Graduate texts in mathematics vol. 82 (1982) (\href{http://www.maths.ed.ac.uk/~aar/papers/botttu.pdf}{pdf})


\item RP Osborne and JL Stern. \emph{Covering Manifolds with Cells}, Pacific Journal of Mathematics, Vol 30, No. 1, 1969.


\item MathOverflow, \emph{\href{http://mathoverflow.net/q/102161/381}{Proving the existence of good covers}}



\end{itemize}
[[!redirects good cover]] [[!redirects good covers]] [[!redirects good open covers]]

[[!redirects good covering]] [[!redirects good coverings]]

[[!redirects differentiably good open cover]] [[!redirects differentiably good open covers]]



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