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\begin{document}

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\section*{partition of unity}

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

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

[[!include topology - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{existence_on_paracompact_topological_spaces}{Existence on paracompact topological spaces}\dotfill \pageref*{existence_on_paracompact_topological_spaces} \linebreak
\noindent\hyperlink{ExistenceOnSmoothManifolds}{Existence on smooth manifolds}\dotfill \pageref*{ExistenceOnSmoothManifolds} \linebreak
\noindent\hyperlink{from_a_nonpoint_finite_partition_of_unity_to_a_partition_of_unity}{From a non-point finite partition of unity to a partition of unity}\dotfill \pageref*{from_a_nonpoint_finite_partition_of_unity_to_a_partition_of_unity} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{maps_to_geometric_realizations}{Maps to geometric realizations}\dotfill \pageref*{maps_to_geometric_realizations} \linebreak
\noindent\hyperlink{CechCoboundaries}{Coboundaries for Cech cocycles}\dotfill \pageref*{CechCoboundaries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{partition of unity} is a [[partition]] of the unit function on a [[topological space]] into a sum of continuous functions that are each non-zero only on small parts of the space.

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

Let $X$ be a [[topological space]]. A (point finite) \textbf{partition of unity} on $X$ is a collection $\{u_j\}_{j \in J}$ of [[continuous functions]] $u_j \colon X \to [0,1]$, $j\in J$ to the [[closed interval]] with its [[Euclidean space|Euclidean]] [[metric topology]] such that

\begin{enumerate}%
\item For each $x\in X$, there is only a [[finite number]] of $j\in J$ such that $u_j(x) \neq 0$ (point finiteness condition);


\item $\sum_{j \in J} u_j(x) = 1$ for all $x\in X$.



\end{enumerate}
A partition of unity defines an [[open cover]] of $X$, consisting of the open sets $u_j^{-1}(0,1]$. Call this the \textbf{induced cover}.

Sometimes (rarely) the condition that $\{u_j\}_J$ is point finite is dropped. In this case we refer to a \emph{non-point finite} partition of unity (see [[red herring principle]]). In this case for each point of $X$ at most countably-many of the functions $u_j$ are non-zero, and we have to interpret the sum in 1. above as being a [[convergence|convergent]] infinite [[series]].

Given a [[cover]] $\mathcal{U} = \{U_j\}_{j\in J}$ of a [[topological space]] ([[open cover]] or closed or neither), the partition of unity $\{u_j\}_J$ is \textbf{subordinate} to $\mathcal{U}$ if for all $j\in J$,

\begin{displaymath}
\overline{u_j^{-1}(0,1]} \subset U_j.
\end{displaymath}
What this means is that the open sets $u_j^{-1}(0,1]$ form an open cover \emph{refining} the cover $\mathcal{U}$.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{example}
\label{PartitionOfUnityOnTheRealLine}\hypertarget{PartitionOfUnityOnTheRealLine}{}
Consider $\mathbb{R}$ with its [[Euclidean space|Euclidean]] [[metric topology]].

Let $\epsilon \in (0,\infty)$ and consider the [[open cover]]

\begin{displaymath}
\{
    (n-1-\epsilon , n+1 +  \epsilon) \subset \mathbb{R}
  \}_{n \in \mathbb{Z} \subset \mathbb{R} }
  \,.
\end{displaymath}
Then a partition of unity $\{ f_n \colon \mathbb{R} \to [0,1] \}_{n \in \mathbb{N}}$ subordinate to this cover is given by

\begin{displaymath}
f_n(x)
    \coloneqq
  \left\{
    \itexarray{
      x - (n - 1)  &\vert& n - 1 \leq x \leq n
      \\
      1- (x-n) &\vert& n \leq x \leq n+1
      \\
      0 &\vert& \text{otherwise}
    }
  \right\}
  \,.
\end{displaymath}
\end{example}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{existence_on_paracompact_topological_spaces}{}\subsubsection*{{Existence on paracompact topological spaces}}\label{existence_on_paracompact_topological_spaces}

\begin{prop}
\label{ParacompactHausdorffEquivalentToexistenceOfParititionsOfUnity}\hypertarget{ParacompactHausdorffEquivalentToexistenceOfParititionsOfUnity}{}
\textbf{([[paracompact Hausdorff spaces equivalently admit subordinate partitions of unity]])}

Assuming the [[axiom of choice]] then:

Let $(X,\tau)$ be a [[topological space]]. Then the following are equivalent:

\begin{enumerate}%
\item $(X,\tau)$ is a [[paracompact Hausdorff space]].


\item Every [[open cover]] of $(X,\tau)$ admits a subordinate partition of unity.



\end{enumerate}
\end{prop}
Similarly [[normal spaces]] are equivalently those such that every [[locally finite cover]] has a subordinate partition of unity (reference Bourbaki, Topology Generale - find this!)

\hypertarget{ExistenceOnSmoothManifolds}{}\subsubsection*{{Existence on smooth manifolds}}\label{ExistenceOnSmoothManifolds}

Paracompact [[smooth manifolds]] even have \emph{smooth} partitions of unity subordinate to any open cover (this follows from the existence of a smooth [[bump function]] on $[-1,1]$). It is not true, however, that [[analytic manifolds]] have analytic partitions of unity - the aforementioned [[bump function]] is smooth but not analytic:

\begin{lemma}
\label{SmoothManifoldClosedBallRefinementOfCover}\hypertarget{SmoothManifoldClosedBallRefinementOfCover}{}
\textbf{([[open cover]] of [[smooth manifold]] admits [[locally finite cover|locally finite]] [[refinement]] by [[closed balls]])}

Let $X$ be a [[smooth manifold]] and let $\{U_i \subset X\}_{i \in I}$ be an [[open cover]]. Then there exists cover

\begin{displaymath}
\left\{
    B_0(\epsilon_j) 
      \underoverset{\simeq}{\psi_j}{\to}
    V_j
      \subset 
    X
  \right\}_{i \in J}
\end{displaymath}
which is a [[locally finite cover|locally finite]] [[refinement]] of $\{U_i \subset X\}_{i \in I}$ with each patch [[diffeomorphism|diffeomorphic]] to a [[closed ball]] in [[Euclidean space]].

\end{lemma}
\begin{proof}
First consider the special case that $X$ is [[compact topological space]].

Let

\begin{displaymath}
\left\{
    \mathbb{R}^n
      \underoverset{\simeq}{\phi_j}{\longrightarrow}
    V_j
    \subset X
  \right\}
\end{displaymath}
be a smooth [[atlas]] representing the [[smooth structure]] on $X$. The [[intersections]]

\begin{displaymath}
\left\{
    U_i \cap V_j
  \right\}_{i \in I, j \in J}
\end{displaymath}
still form an open cover of $X$. Hence for each point $x \in X$ there is $i \in I$ and $j \in J$ with $x \in U_i \cap V_j$. By the nature of the [[Euclidean topology]], there exists a [[closed ball]] $B_x$ around $\phi_j^{-1}(x)$ in $\phi_j^{-1}(U_i \cap V_j) \subset \mathbb{R}^n$. Its [[image]] $\phi_j(B_x) \subset X$ is a neighbourhood of $x \in X$ diffeomorphic to a closed ball.

The [[interiors]] of these balls form an [[open cover]]

\begin{displaymath}
\left\{
    Int(B_x)
     \subset 
    X
  \right\}_{x \in X}
\end{displaymath}
of $X$ which, by construction, is a refinement of $\{U_i \subset X\}_{i \in I}$. By the assumption that $X$ is compact, this has a finite subcover

\begin{displaymath}
\left\{
    Int(B_l)
    \subset 
    X
  \right\}_{l \in L}
\end{displaymath}
for $L$ a [[finite set]]. Hence

\begin{displaymath}
\left\{
    B_l
    \subset 
    X
  \right\}_{l \in L}
\end{displaymath}
is a finite cover by closed balls, hence in particular locally finite, and by construction it is still a refinement of the orignal cover. This shows the statement for $X$ compact.

Now for general $X$, notice that without restriction we may assume that $X$ is [[connected topological space|connected]], for if it is not, then we obtain the required refinement on all of $X$ by finding one on each [[connected component]].

But if a locally Euclidean paracompact Hausdorff space $X$ is connected, then it is [[sigma-compact topological space|sigma-compact]] and in fact admits a countable increasing exhaustion

\begin{displaymath}
V_0 \subset V_1 \subset V_2 \subset \cdots
\end{displaymath}
by [[open subsets]] whose [[topological closures]]

\begin{displaymath}
K_0 \subset K_1 \subset K_2 \subset \cdots
\end{displaymath}
exhaust $X$ by [[compact topological space|compact]] subspaces $K_n$ (by the proof of \href{topological+manifold#RegularityConditionsForTopologicalManifoldsComparison}{this prop.}).

For $n \in \mathbb{N}$, consider the open subspace

\begin{displaymath}
V_{n+2} \setminus K_{n-1}
    \;\subset\;
  X
\end{displaymath}
which canonically inherits the structure of a smooth manifold (\href{differentiable+manifold#OpenSubsetsOfDifferentiableManifoldsAreDifferentiableManifolds}{this prop.}). As above we find a refinement of the restriction of $\{U_i \subset X\}_{i \in I}$ to this open subset by closed balls and since the further subspace $K_{n+1}\setminus K_n$ is still compact (by \href{compact+space#UnionsAndIntersectionOfCompactSubspaces}{this lemma}) there is a finite set $L_n$ such that

\begin{displaymath}
\{B_{l_n} \subset V_{n+2} \setminus K_{n-1} \subset X \}_{l_n \in L_n}
\end{displaymath}
is a finite cover of $K_{n+1} \setminus K_n$ by closed balls refining the original cover.

It follows that the union of all these

\begin{displaymath}
\left\{
   B_{l_n} \subset X
  \right\}_{n \in \mathbb{N}, l_n \in L_n}
\end{displaymath}
is a refinement by closed balls as required. Its local finiteness follows by the fact that each $B_{l_n}$ is contained in the ``strip'' $V_{n+2} \setminus K_{n-1}$, each strip contains only a finite set of $B_{l_n}$-s and each strip intersects only a finite number of other strips. (Hence an open subset around a point $x$ which intersects only a finite number of elements of the refined cover is given by any one of the balls $B_{l_n}$ that contain $x$.)

\end{proof}
\begin{prop}
\label{SmoothManifoldAdmitsSmoothPartitionsOfUnity}\hypertarget{SmoothManifoldAdmitsSmoothPartitionsOfUnity}{}
\textbf{([[smooth manifolds]] admit smooth partitions of unity)}

Let $X$ be a paracompact [[smooth manifold]]. Then every [[open cover]] $\{U_i \subset X\}_{i \in I}$ has a subordinate partition of unity by functions $\{f_i \colon U_i \to \mathbb{R}\}_{i \in I}$ which are \emph{[[smooth functions]]}.

\end{prop}
\begin{proof}
By lemma \ref{SmoothManifoldClosedBallRefinementOfCover} the given cover has a [[locally finite cover|locally finite]] [[refinement]] by [[closed subsets]] [[diffeomorphism|diffeomorphic]] to [[closed balls]]:

\begin{displaymath}
\left\{
    B_0(\epsilon_j) 
      \underoverset{\simeq}{\psi_j}{\to}
    V_j
      \subset 
    X
  \right\}_{j \in J}
  \,.
\end{displaymath}
Given this, let

\begin{displaymath}
h_j \;\colon\; X \longrightarrow \mathbb{R}
\end{displaymath}
be the function which on $V_j$ is given by a smooth [[bump function]]

\begin{displaymath}
b_j \;\colon\; \mathbb{R} \longrightarrow \mathbb{R}
\end{displaymath}
with [[support]] $supp(b_j) = B_0(\epsilon_j)$:

\begin{displaymath}
h_j
  \;\colon\;
  x 
  \mapsto
  \left\{
    \itexarray{
      b_j(\psi_j^{-1}(x)) &\vert& x \in V_j
      \\
      0 &\vert& \text{otherwise}
    }
  \right.
  \,.
\end{displaymath}
By the nature of [[bump functions]] this is indeed a [[smooth function]] on all of $X$. By local finiteness of the cover by closed balls, the function

\begin{displaymath}
h \;\colon\; X \longrightarrow \mathbb{R}
\end{displaymath}
given by

\begin{displaymath}
h(x) 
    \coloneqq 
  \underset{j \in J}{\sum} h_j(x)
\end{displaymath}
is well defined (the sum involves only a finite number of non-vanishing contributions) and is smooth. Therefore setting

\begin{displaymath}
f_j \;\coloneqq\; \frac{h_j}{h}
\end{displaymath}
then

\begin{displaymath}
\left\{
    f_j
 \right\}_{j \in J}
\end{displaymath}
is a subordinate partition of unity by smooth functions as required.

\end{proof}
\hypertarget{from_a_nonpoint_finite_partition_of_unity_to_a_partition_of_unity}{}\subsubsection*{{From a non-point finite partition of unity to a partition of unity}}\label{from_a_nonpoint_finite_partition_of_unity_to_a_partition_of_unity}

\begin{defn}
\label{}\hypertarget{}{}
A collection of functions $\mathcal{U} = \{u_i : X \to [0,1]\}$ such that every $x\in X$ is in the support of some $u_i$. Then $\mathcal{U}$ is called \emph{locally finite} if the cover $u_i^{-1}(0,1]$ (i.e. the induced cover) is [[locally finite cover|locally finite]].

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
(Mather, 1965)

Let $\{u_i\}_J$ be a non-point finite partition of unity. Then there is a locally finite partition of unity $\{v_i\}_{i\in J}$ such that the induced cover of the latter is a refinement of the induced cover of the former.

\end{prop}
(For a proof, see p.354 of Dold's Lectures on algebraic topology. \href{https://books.google.com.au/books?id=4AmzD9XhGDkC&lpg=PA354&vq=locally%20finite%20oartition%20unity&pg=PA354#v=onepage&q&f=false}{Google books link to page 354}, which may or may not be visible)

This implies that (loc. finite) [[numerable covers]] are cofinal in induced covers arising from collections of functions as in the definition. In particular, given the [[Milnor classifying space]] $\mathcal{B}^M G$ of a [[topological group]] $G$, which comes with a countable family of `coordinate functions' $\mathcal{B}^M G \to [0,1]$, has a numerable cover. This is shown by Dold to be a trivialising cover for the universal bundle constructed by Milnor, and so the universal bundle is [[numerable bundle|numerable]].

\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\hypertarget{maps_to_geometric_realizations}{}\subsubsection*{{Maps to geometric realizations}}\label{maps_to_geometric_realizations}

Partitions of unity can be used in constructing maps from spaces to [[geometric realization]]s of [[simplicial spaces]] (incl. simplicial sets) - for example a [[classifying map]] for a $G$-[[bundle]] where $G$ is a [[Lie group]].

\hypertarget{CechCoboundaries}{}\subsubsection*{{Coboundaries for Cech cocycles}}\label{CechCoboundaries}

Partitions of unity can be used to give explicit coboundaries for the cocycles of the complex of functions on a cover.

Let $\{U_i \to X\}$ be a [[open cover]] and $\{\rho_i \in C(X,\mathbb{R})\}$ a collection of functions with

\begin{itemize}%
\item $(x not \in U_i) \Rightarrow \rho_i(x) = 0$


\item $\sum_i \rho_i = const_1$.



\end{itemize}
Write $C(\{U_i\}) : \Delta^{op} \to Top$ for the [[Cech nerve]] of the cover and $C(C(\{U_i\}), \mathbb{R})$ for the [[cosimplicial ring]] of functions on this [[simplicial object|simplicial]] topological space; and $(C_\bullet(C(\{U_i\}), \mathbb{R}), \delta)$ for the corresponding (normalized) [[Moore complex|cochain complex]]: its differential is the alternating sum of the pullbacks of functions along the face maps, i.e. along the restriction maps

\begin{displaymath}
\delta = \sum_k (-1)^k \delta_{k}^*
  \,.
\end{displaymath}
For instance for $f = \{f_{i_1, i_2, \cdots, i_n} \in C(U_{i_1} \cap \cdots \cap U_{i_{n+1}})\}$ a collection of functions in degree $n$, we have

\begin{displaymath}
(\delta f)_{i_0 \cdots i_n i_{n+1}} =
  \sum_{k = 0}^{n+1} (-1)^k f_{i_0 \cdots i_{k-1} i_{k+1} \cdots i_{n+1}}
  \,.
\end{displaymath}
This cochain complex has vanishing [[cochain cohomology]] in positive degree. We can explicitly construct corresponding coboundaries using the partion of unity:

assume that with the above notation $f$ is a cocycle in positive degree, in that $\delta f  = 0$. Then define the $(n-1)$-cochain

\begin{displaymath}
\lambda_{i_1 \cdots i_n}
  :=
  \sum_{i_0} \rho_{i_0} f_{i_0 i_1 \cdots i_n}
  \,.
\end{displaymath}
Here in the summands on the right the product is defined on $U_{i_0} \cap U_{i_1} \cap \cdots \cap U_{i_n}$ and extended as 0 to all of $U_{i_1} \cap \cdots \cap U_{i_n}$.

With this definition we have

\begin{displaymath}
\delta \lambda = f
  \,.
\end{displaymath}
To see this we compute

\begin{displaymath}
\begin{aligned}
    (\delta \lambda)_{i_1 \cdots i_{n+1}}
    & :=
    \sum_{i_0} \rho_{i_0} \sum_{k=1}^n (-1)^k f_{i_0 i_1 \cdots i_{k-1} i_{k+1} \cdots i_{n+1}}
    \\
    & =
    \pm 
    \sum_{i_0} \rho_{i_0} 
    f_{i_1 \cdots i_{n+1}}
    \\
    & = f_{i_1 \cdots i_{n+1}}
  \end{aligned}
  \,,
\end{displaymath}
where in the second step we used the condition $\delta f = 0$ and in the last step we used the property of the partition of unity.

This construction is used a lot in [[Cech cohomology]]. For instance it can be used to show in Chech cocycles that every [[principal bundle]] admits a [[connection on a bundle]] (see there for the details).

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

\begin{itemize}%
\item [[Albrecht Dold]], \emph{Partitions of unity in the theory of fibrations}, Ann. of Math. 78. (1963), 223-255.


\item [[Albrecht Dold]], \emph{Lectures on algebraic topology}, Springer Classics in Mathematics (1980), p.354.


\item M. Mather, \emph{Paracompactness and partitions of unity}, PhD thesis, Cambridge (1965).



\end{itemize}
Discussion of partitions of unity in [[constructive mathematics]] is in

\begin{itemize}%
\item [[Frank Waaldijk]], section 3.1 of \emph{modern intuitionistic topology}, 1996 (\href{http://www.fwaaldijk.nl/modern%20intuitionistic%20topology.pdf}{pdf})

\end{itemize}
[[!redirects partition of unity]] [[!redirects partitions of unity]] [[!redirects Partition of unity]] [[!redirects Partitions of unity]]



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