# Differentiable Manifolds/Maximal atlases, second-countable spaces and partitions of unity

 Differentiable Manifolds ← Bases of tangent and cotangent spaces and the differentials Maximal atlases, second-countable spaces and partitions of unity Submanifolds →

## Maximal atlases

Definition 3.1:

Let ${\displaystyle M}$ be a ${\displaystyle d}$-dimensional manifold of class ${\displaystyle {\mathcal {C}}^{n}}$ and let ${\displaystyle \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}}$ be it's atlas. We call the set

${\displaystyle A:=\left\{(U,\theta )|U\subseteq M{\text{ open }},\theta :U\to \mathbb {R} ^{d},\theta {\text{ is compatible of class }}{\mathcal {C}}^{n}{\text{ to all }}\phi _{\upsilon },\upsilon \in \Upsilon \}\right\}}$

the maximal atlas of ${\displaystyle M}$.

Lemma 3.2: We have ${\displaystyle \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}\subseteq A}$.

Proof: This is because if ${\displaystyle (O,\phi )\in \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}}$, then by definition of an atlas it is compatible with all the elements of ${\displaystyle \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}}$ and hence, by definition of ${\displaystyle A}$, contained in ${\displaystyle A}$.${\displaystyle \Box }$

Theorem 3.3: The maximal atlas really is an atlas; i. e. for every point ${\displaystyle p\in M}$ there exists ${\displaystyle (U,\theta )}$ such that ${\displaystyle p\in M}$, and every two charts in it are compatible.

Proof:

1.

We first show that for every point ${\displaystyle p\in M}$ there exists ${\displaystyle (U,\theta )}$ such that ${\displaystyle p\in M}$:

From lemma 3.2 we know that the atlas of ${\displaystyle M}$ is contained in ${\displaystyle A}$.

Let now ${\displaystyle p\in M}$. Due to the definition of an atlas, we find an ${\displaystyle (U,\theta )\in \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}}$ such that ${\displaystyle p\in O}$. Since ${\displaystyle \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}\subseteq A}$, we obtain ${\displaystyle (U,\theta )\in A}$.

2.

We prove that every two charts ${\displaystyle \phi :O\to \mathbb {R} ^{d},\theta :U\to \mathbb {R} ^{d}}$ such that ${\displaystyle (O,\phi ),(U,\theta )\in A}$, are compatible.

So let ${\displaystyle \phi :O\to \mathbb {R} ^{d},\theta :U\to \mathbb {R} ^{d}}$ such that ${\displaystyle (O,\phi ),(U,\theta )\in A}$ be 'arbitrary' (of course we still require ${\displaystyle (O,\phi ),(U,\theta )\in A}$).

If we have ${\displaystyle O\cap U=\emptyset }$, this directly implies compatibility (recall that we defined compatibility so that if ${\displaystyle U\cap O=\emptyset }$ for two charts ${\displaystyle \phi :O\to \mathbb {R} ^{d},\theta :U\to \mathbb {R} ^{d}}$, then the two are by definition automatically compatible).

So in this case, we are finished. Now we shall prove the other case, which namely is ${\displaystyle O\cap U\neq \emptyset }$.

Due to the definition of compatibility of class ${\displaystyle {\mathcal {C}}^{n}}$, we have to prove that the function

${\displaystyle \phi |_{U\cap O}\circ \psi |_{U\cap O}^{-1}:\psi (U\cap O)\to \phi (U\cap O)}$

is contained in ${\displaystyle {\mathcal {C}}^{n}(\psi (U\cap O),\mathbb {R} ^{d})}$ and

${\displaystyle \psi |_{U\cap O}\circ \phi |_{U\cap O}^{-1}:\phi (U\cap O)\to \psi (U\cap O)}$

is contained in ${\displaystyle {\mathcal {C}}^{n}(\phi (U\cap O),\mathbb {R} ^{d})}$.

Let ${\displaystyle p\in O\cap U}$. Since ${\displaystyle \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}}$ is the atlas of ${\displaystyle M}$, we find a chart ${\displaystyle (\chi ,V)\in \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}}$ such that ${\displaystyle p\in V}$. Due to the definition of ${\displaystyle A}$, ${\displaystyle \chi }$ and ${\displaystyle \psi }$ are compatible and ${\displaystyle \chi }$ and ${\displaystyle \phi }$ are compatible. Hence, the functions

${\displaystyle \phi |_{V\cap U\cap O}\circ \chi |_{V\cap U\cap O}^{-1}\circ \chi |_{V\cap U\cap O}\circ \psi |_{V\cap U\cap O}^{-1}:\psi (V\cap U\cap O)\to \phi (V\cap U\cap O)}$

and

${\displaystyle \psi |_{V\cap U\cap O}\circ \chi |_{V\cap U\cap O}^{-1}\circ \chi |_{V\cap U\cap O}\circ \phi |_{V\cap U\cap O}^{-1}:\psi (V\cap U\cap O)\to \phi (V\cap U\cap O)}$

are ${\displaystyle n}$-times differentiable (or, if ${\displaystyle n=0}$, continuous), in particular at ${\displaystyle \psi (p)}$, ${\displaystyle \phi (p)}$ respectively. Since ${\displaystyle p\in O\cap U}$ was arbitrary, since

${\displaystyle \phi |_{V\cap U\cap O}\circ \chi |_{V\cap U\cap O}^{-1}\circ \chi |_{V\cap U\cap O}\circ \psi |_{V\cap U\cap O}^{-1}=(\phi |_{U\cap O}\circ \psi |_{U\cap O}^{-1})|_{\psi (V\cap U\cap O)}}$

and

${\displaystyle \psi |_{V\cap U\cap O}\circ \chi |_{V\cap U\cap O}^{-1}\circ \chi |_{V\cap U\cap O}\circ \phi |_{V\cap U\cap O}^{-1}=(\psi |_{U\cap O}\circ \phi |_{U\cap O}^{-1})|_{\phi (V\cap U\cap O)}}$

(which you can show by direct calculation!) and since ${\displaystyle \phi ,\psi }$ are bijective, this shows the theorem.${\displaystyle \Box }$

Theorem 3.4:

Let ${\displaystyle M}$ be a ${\displaystyle d}$-dimensional manifold with atlas ${\displaystyle \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}}$, and let ${\displaystyle A}$ be it's maximal atlas. There does not exist an atlas ${\displaystyle B}$ such that ${\displaystyle A\subsetneq B}$ (this notation shall mean that ${\displaystyle A}$ is contained in ${\displaystyle B}$, but ${\displaystyle B}$ is 'strictly larger than ${\displaystyle A}$' (by this obscure saying we shall mean that there exists at least one element in ${\displaystyle B}$ which is not contained in ${\displaystyle A}$)).

This is, in fact, the reason why the word maximal atlas for ${\displaystyle A}$ does not completely miss the point.

Proof: We show that there does not exist an atlas ${\displaystyle B}$ of ${\displaystyle M}$ such that ${\displaystyle A\subsetneq B}$.

Assume by contradiction that there exists such an atlas. Then we find an element ${\displaystyle (U,\theta )\in B\setminus A}$. But since ${\displaystyle B}$ is an atlas, ${\displaystyle \theta }$ is compatible to all other charts ${\displaystyle \phi }$ for which ${\displaystyle (O,\phi )\in A}$. This means, due to lemma 3.2, that it is compatible to every ${\displaystyle \phi _{\upsilon },\upsilon \in \Upsilon }$. Hence, due to the definition of ${\displaystyle A}$, ${\displaystyle (U,\theta )\in A}$. This is a contradiction!${\displaystyle \Box }$

## Second-countable spaces

Definition 3.5:

Let ${\displaystyle M}$ be a topological space and let ${\displaystyle U_{\kappa },\kappa \in \mathrm {K} }$ be a set of open sets. We call ${\displaystyle \{U_{\kappa }|\kappa \in \mathrm {K} \}}$ a basis of the topology of ${\displaystyle M}$ iff every open set ${\displaystyle O\subseteq M}$ can be written as the union of elements of ${\displaystyle \{U_{\kappa }|\kappa \in \mathrm {K} \}}$, i. e.

${\displaystyle O=\bigcup _{\upsilon \in \Upsilon }U_{\upsilon }}$

where ${\displaystyle \Upsilon \subseteq \mathrm {K} }$.

Definition 3.6:

Let ${\displaystyle M}$ be a topological space. We call ${\displaystyle M}$ second-countable iff ${\displaystyle M}$'s topology has a countable basis.

## Locally finite refinements and partitions of unity

Definition 3.7:

Let ${\displaystyle {\mathcal {X}}}$ be a topological space. An open cover of ${\displaystyle {\mathcal {X}}}$ is a set ${\displaystyle \{V_{\kappa }|\kappa \in \mathrm {K} \}}$ of open subsets of ${\displaystyle {\mathcal {X}}}$ such that

${\displaystyle \bigcup _{\kappa \in \mathrm {K} }V_{\kappa }={\mathcal {X}}}$

Example 3.8:

The set ${\displaystyle \{B_{\epsilon }(x)|x\in \mathbb {R} ^{d},\epsilon >0\}}$ is an open cover of the real numbers.

Definition 3.9:

Let ${\displaystyle M}$ be a manifold of class ${\displaystyle {\mathcal {C}}^{k}}$. We say that ${\displaystyle M}$ admits partitions of unity iff for every open cover ${\displaystyle \{V_{\kappa }|\kappa \in \mathrm {K} \}}$ there exist functions ${\displaystyle \{\varphi _{\iota }|\iota \in \mathrm {I} \}\subset {\mathcal {C}}^{n}(M)}$ such that:

1. For all ${\displaystyle \iota \in \mathrm {I} }$, there exists a ${\displaystyle \kappa \in \mathrm {K} }$ such that ${\displaystyle {\text{supp }}\varphi _{\iota }\subset V_{\kappa }}$,
2. for all ${\displaystyle \iota \in \mathrm {I} }$ and ${\displaystyle p\in M}$, ${\displaystyle \varphi _{\iota }(p)\geq 0}$ AND
3. for all ${\displaystyle p\in M}$, ${\displaystyle \sum _{\iota \in \mathrm {I} }\varphi _{\iota }(p)=1}$.

Definition 3.10:

Let ${\displaystyle {\mathcal {X}}}$ be a topological space and let ${\displaystyle \{V_{\kappa }|\kappa \in \mathrm {K} \}}$ be an open cover of ${\displaystyle {\mathcal {X}}}$. A locally finite refinement of ${\displaystyle \{V_{\kappa }|\kappa \in \mathrm {K} \}}$ is defined to be another open cover of ${\displaystyle {\mathcal {X}}}$, say ${\displaystyle \{U_{\upsilon }|\upsilon \in \Upsilon \}}$, such that:

• for each ${\displaystyle \upsilon \in \Upsilon }$, there exists a ${\displaystyle \kappa \in \mathrm {K} }$ such that ${\displaystyle U_{\upsilon }\subseteq V_{\kappa }}$, AND
• for each ${\displaystyle x\in {\mathcal {X}}}$, the set ${\displaystyle \{U_{\upsilon }|\upsilon \in \Upsilon \wedge x\in V_{\upsilon }\}}$ is finite.

We will now prove a few lemmas, which will help us to prove that every manifold whose topology has a countable basis admits partition of unity. Then, we will prove that every manifold whose topology has a countable basis admits partition of unity :-)

Lemma 3.11:

Let ${\displaystyle M}$ be a manifold with a countable basis. Then ${\displaystyle M}$ has a countable basis ${\displaystyle \{V_{j}|j\in \mathbb {N} \}}$ such that for each ${\displaystyle j\in \mathbb {N} }$, ${\displaystyle {\overline {U_{j}}}}$ is compact.

Proof:

Let ${\displaystyle \{U_{k}|k\in \mathbb {N} \}}$ be a countable basis of ${\displaystyle M}$. For each ${\displaystyle p\in M}$, we choose a chart ${\displaystyle (O,\phi )}$ such that ${\displaystyle p\in O}$. Then we choose ${\displaystyle W_{p}:=\phi (O)\cap B_{1}(\phi (p))}$. Since in ${\displaystyle \mathbb {R} ^{d}}$, sets are compact if and only if bounded and closed, ${\displaystyle {\overline {W_{p}}}}$ is compact. There is a theorem from topology, which states that the image of a compact set under a homeomorphism is again compact. Hence, ${\displaystyle \phi ^{-1}({\overline {W_{p}}})}$ is a compact subset of ${\displaystyle O}$.

Further, if ${\displaystyle \{V_{\kappa },\kappa \in \mathrm {K} \}}$ is an cover of ${\displaystyle \phi ^{-1}({\overline {W_{p}}})}$ by open subsets of ${\displaystyle M}$, then the set ${\displaystyle \{V_{\kappa }\cap O,\kappa \in \mathrm {K} \}}$ is a cover of ${\displaystyle \phi ^{-1}({\overline {W_{p}}})}$ by open subsets of ${\displaystyle O}$. Since ${\displaystyle \phi ^{-1}({\overline {W_{p}}})}$ is compact in ${\displaystyle O}$, we may pick out of the latter a finite subcover ${\displaystyle V_{\kappa _{1}}\cap O,\ldots ,V_{\kappa _{n}}\cap O}$. Then, since

${\displaystyle O\subseteq \bigcup _{j=1}^{n}V_{\kappa _{j}}\cap O\subseteq \subseteq \bigcup _{j=1}^{n}V_{\kappa _{j}}}$

, the set ${\displaystyle V_{\kappa _{1}},\ldots ,V_{\kappa _{n}}}$ is a finite subcover of ${\displaystyle \{V_{\kappa },\kappa \in \mathrm {K} \}}$. Thus, ${\displaystyle \phi ^{-1}({\overline {W_{p}}})}$ is also a compact subset of ${\displaystyle M}$.

As ${\displaystyle \phi }$ is a homeomorphism, ${\displaystyle W_{p}}$ is open in ${\displaystyle M}$, and from ${\displaystyle W_{p}\subset {\overline {W_{p}}}}$, it follows ${\displaystyle \phi ^{-1}(W_{p})\subset \phi ^{-1}({\overline {W_{p}}})}$. Thus, also

${\displaystyle {\overline {\phi ^{-1}(W_{p})}}\subseteq \phi ^{-1}({\overline {W_{p}}})}$

since the closure of ${\displaystyle \phi ^{-1}(W_{p})}$ is, by definition (with the definition of some lectures), equal to

${\displaystyle \bigcap _{A\supseteq \phi ^{-1}(W_{p}) \atop A{\text{ closed }}}A}$

Further, another theorem from topology states that closed subsets of compact sets are compact. Hence, ${\displaystyle {\overline {\phi ^{-1}(W_{p})}}}$ is compact.

Since ${\displaystyle \{U_{k}|k\in \mathbb {N} \}}$ was a basis, each of the ${\displaystyle \phi ^{-1}(W_{p})}$ can be written as the union of elements of ${\displaystyle \{U_{k}|k\in \mathbb {N} \}}$. We choose now our new basis as consisting of the union over ${\displaystyle p\in M}$ of the elements of ${\displaystyle \{U_{k}|k\in \mathbb {N} \}}$ with smallest index ${\displaystyle m_{p}}$, such that ${\displaystyle p\in U_{m_{p}}}$ and ${\displaystyle U_{m_{p}}\subseteq \phi ^{-1}(W_{p})}$. Now the closures of the ${\displaystyle U_{m_{p}}}$ are compact: From ${\displaystyle U_{m_{p}}\subseteq \phi ^{-1}(W_{p})}$ follows that ${\displaystyle {\overline {U_{m_{p}}}}\subseteq {\overline {\phi ^{-1}(W_{p})}}}$, and since ${\displaystyle {\overline {\phi ^{-1}(W_{p})}}\subseteq \phi ^{-1}({\overline {W_{p}}})}$, ${\displaystyle {\overline {U_{m_{p}}}}}$ is compact as the closed subset of a compact set.

Since our new basis is a subset of a countable set, it is itself countable (we include finite sets in the category 'countable' here). Thus, we have obtained a countable basis the elements of which have compact closure.${\displaystyle \Box }$

Lemma 3.12:

Let ${\displaystyle M}$ be a manifold with a countable base (i. e. a second-countable manifold). Then for every cover of ${\displaystyle M}$ there is a locally finite refinement.

Proof:

Let ${\displaystyle \{V_{\kappa }|\kappa \in \mathrm {K} \}}$ be a cover of ${\displaystyle M}$. Due to lemma 3.11, we may choose a countable basis ${\displaystyle \{U_{j}|j\in \mathbb {N} \}}$ of ${\displaystyle M}$ such that each ${\displaystyle {\overline {U_{j}}}}$ is compact. We now define a sequence of compact sets ${\displaystyle (A_{l})_{l\in \mathbb {N} }}$ inductively as follows: We set ${\displaystyle A_{1}:={\overline {U_{1}}}}$. Once we defined ${\displaystyle A_{l}}$, we define

${\displaystyle A_{l+1}:={\overline {U_{1}}}\cup \cdots \cup {\overline {U_{m}}}}$

, where ${\displaystyle m\in \mathbb {N} }$ is smallest such that we have:

${\displaystyle {\text{int }}A_{l}\subset {\overline {U_{1}}}\cup \cdots \cup {\overline {U_{m}}}}$

This is compact, since a theorem from topology states that the finite union of compact sets is compact. Since, as mentioned before, there is a theorem from topology stating that closed subsets of compact sets are compact, the sets defined by ${\displaystyle B_{1}:=A_{1}}$ and

${\displaystyle B_{l}:=A_{l}\setminus {\text{int }}A_{l-1}}$

for ${\displaystyle l\geq 2}$ (intuitively the closed annulus) are compact. Further, the sets ${\displaystyle C_{l}}$, defined by ${\displaystyle C_{1}:={\text{int }}A_{2}}$, ${\displaystyle C_{2}:={\text{int }}A_{3}}$ and

${\displaystyle C_{l}:={\text{int }}A_{l+1}\setminus A_{l-2}}$

for ${\displaystyle l\geq 3}$ (intuitively the next bigger open annulus) are open, and we have for all ${\displaystyle l\in \mathbb {N} }$:

${\displaystyle B_{l}\subset C_{l}}$

Now since ${\displaystyle M}$ is covered by ${\displaystyle \{V_{\kappa }|\kappa \in \mathrm {K} \}}$, so is each of the sets ${\displaystyle B_{l}}$. Now we compose our locally finite refinement as follows: We include all the sets, which are the intersection of ${\displaystyle C_{l}}$ and the (by compactness existing) sets of the finite subcovers of ${\displaystyle B_{l}}$ out of ${\displaystyle \{V_{\kappa }|\kappa \in \mathrm {K} \}}$. This is a locally finite refinement.${\displaystyle \Box }$

Lemma 3.13:

Let ${\displaystyle M}$ be a ${\displaystyle d}$-dimensional manifold of class ${\displaystyle {\mathcal {C}}^{n}}$ with atlas ${\displaystyle \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}}$, let ${\displaystyle (O,\phi )\in \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}}$, let ${\displaystyle W\subseteq O}$ be open in ${\displaystyle O}$ (with respect to the subspace topology and let ${\displaystyle p\in O}$ and let ${\displaystyle \epsilon >0}$ be such that ${\displaystyle B_{\epsilon }(\phi (p))\subseteq \phi (O\cap W)}$. If we define

${\displaystyle \eta :\mathbb {R} ^{d}\to \mathbb {R} ,\eta (x)={\begin{cases}e^{1-{\frac {1}{1-\|x\|^{2}}}}&{\text{ if }}\|x\|_{2}<1\\0&{\text{ if }}\|x\|_{2}\geq 1\end{cases}}}$

, and

${\displaystyle h_{p,W,\phi }:M\to \mathbb {R} ,h_{p,W,\phi }(q):={\begin{cases}\eta \left({\frac {1}{\epsilon }}(\phi (p)-\phi (q))\right)&q\in O\\0&q\notin O\\\end{cases}}}$,

then we have ${\displaystyle h_{p,W,\phi }\in {\mathcal {C}}^{n}(M)}$.

Proof:

Let ${\displaystyle (U,\theta )\in \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}}$. Then we have for ${\displaystyle x\in \theta (U)}$:

${\displaystyle (h_{p,W,\phi }|_{U}\circ \theta ^{-1})(x)={\begin{cases}0&(\phi \circ \theta ^{-1})(x)\notin B_{\epsilon }(\phi (p))\\e^{1-{\frac {1}{1-\|{\frac {1}{\epsilon }}\left(\phi (p)-(\phi \circ \theta ^{-1})(x)\right)\|^{2}}}}&(\phi \circ \theta ^{-1})(x)\in B_{\epsilon }(\phi (p))\end{cases}}}$

This function is ${\displaystyle n}$ times differentiable (or continuous if ${\displaystyle n=0}$) as the composition of ${\displaystyle n}$ times differentiable (or continuous if ${\displaystyle n=0}$) functions.${\displaystyle \Box }$

Theorem 3.14:

Let ${\displaystyle M}$ be a manifold of class ${\displaystyle {\mathcal {C}}^{n}}$ with a countable basis, i. e. a second-countable manifold. Then ${\displaystyle M}$ admits partitions of unity.

Proof:

Let ${\displaystyle \{V_{\kappa }|\kappa \in \mathrm {K} \}}$ be an open cover of ${\displaystyle M}$.

We choose for each point ${\displaystyle p\in M}$ an atlas ${\displaystyle (O,\phi )}$ such that ${\displaystyle p\in O}$. Further, we choose an arbitrary ${\displaystyle V_{\kappa _{p}}}$ in the open cover such that ${\displaystyle p\in V_{\kappa _{p}}}$. By definition of the subspace topology we have that ${\displaystyle V_{\kappa _{p}}\cap O}$ is open in ${\displaystyle O}$. Therefore, due to lemma 3.13, we may choose ${\displaystyle h_{p,V_{\kappa _{p}}\cap O,\phi }\in {\mathcal {C}}^{n}(M)}$ such that ${\displaystyle p\in \{q\in M|h_{p,V_{\kappa _{p}}\cap O,\phi }(q)>0\}=:W_{p}}$. Since ${\displaystyle h_{p,V_{\kappa _{p}}\cap O,\phi }}$ is continuous, all the ${\displaystyle W_{p}}$ are open; this is because they are preimages of the open set ${\displaystyle (0,\infty )}$. Further, since there is a ${\displaystyle W_{p}}$ for every ${\displaystyle p\in M}$, and always ${\displaystyle p\in W_{p}}$, the ${\displaystyle W_{p}}$ form a cover of ${\displaystyle M}$. Due to lemma 3.12 we may choose a locally finite refinement. This open cover, this set of open sets we shall denote by ${\displaystyle S}$.

We now define the function

${\displaystyle \varphi :M\to \mathbb {R} ,\varphi (q):=\sum _{W_{p}\in S}h_{p,V_{\kappa _{p}}\cap O,\phi }(q)}$

This function is of class ${\displaystyle {\mathcal {C}}^{n}(M)}$ as a finite sum (because for each ${\displaystyle p}$ there are only finitely many ${\displaystyle W\in S}$ such that ${\displaystyle p\in W}$, because ${\displaystyle S}$ was a locally finite subcover) of ${\displaystyle {\mathcal {C}}^{n}(M)}$ functions (that finite sums of ${\displaystyle {\mathcal {C}}^{n}(M)}$ functions are again ${\displaystyle {\mathcal {C}}^{n}(M)}$ follows from theorem 2.22 and induction) and does not vanish anywhere (since for every ${\displaystyle p}$ there is a ${\displaystyle W_{q}\in S}$ such that ${\displaystyle p}$ is in it; remember that a finite refinement is an open cover), and therefore follows from theorem 2.26, that all the functions ${\displaystyle \varphi _{p,V_{\kappa _{p}}\cap O,\phi }:={\frac {h_{p,V_{\kappa _{p}}\cap O,\phi }}{\varphi }}}$ are contained in ${\displaystyle {\mathcal {C}}^{n}(M)}$. It is not difficult to show that these functions are non-negative and that they sum up to ${\displaystyle 1}$ at every point. Further due to the construction, each of their supports is contained in one ${\displaystyle V_{\kappa }}$. Thus they form the desired partition of unity.${\displaystyle \Box }$

## Sources

• Lang, Serge (2002). Introduction to Differentiable Manifolds. New York: Springer. ISBN 0-387-95477-5.
 Differentiable Manifolds ← Bases of tangent and cotangent spaces and the differentials Maximal atlases, second-countable spaces and partitions of unity Submanifolds →