4. Paracompactness

4.1Compact and Paracompact

定义 4.1.1. A topological space $(x,\mathcal{O})$ is compact if every open covering has a finite subcover.

例 4.1.2.  

•	Compact $\{x\}$, $[0,1]$, $S^1$, $S^n$, $T^n$.
•	Not compact $(0,1)$, $(0,1]$, $\mathbb{R}$, ${\mathbb{R}}^n$.

定义 4.1.3. A topological space $(X,\mathcal{O})$ is paracompact if every open covering has a locally finite refinement.

定义 4.1.4. Let $\{U_i\mid i\in I\}$, be a collection of subsets in $X$. This collection is locally finite if $\forall x\in X$, there exists an open neighborhood $U_x$, s.t. $U_i\cap U_x\ne \varnothing$ for only finitely many $i\in I$.

定义 4.1.5. Let $U_i$, $i\in I$ be a covering of $X$, i.e. $\bigcup _{i\in I}{U}_i=X$. A refinement of this covering is a covering by subsets $V_k$, $k\in K$, such that $\forall k\in K$, $\exists i=i(k)\in I$, s.t. $V_k\subset U_i$.

命题 4.1.6. Let $\{U_i\mid i\in I\}$ be an open covering of a manifold $M$. There exists an atlas $\mathcal{A}=\{(V_k,{\phi }_k)\mid k\in K\}$ such that

(1)	${\phi }_k(V_k)=B(x_k,3)\subset {\mathbb{R}}^n$;
(2)	$W_k={\phi }_k^{-1}(B(x_k,1))$ form a covering of $M$;
(3)	The $V_k$ form a locally finite refinement of the covering by the $U_i$.

证明. Step 1: There exists a sequence $G_i$, $i=1,2,\dots$ of open subsets on $M$ with ${\overline{G}}_i\subset G_{i+1}$ $\forall i$, ${\overline{G}}_i$ compact $\forall i$, and $\bigcup _{i=1}^{\infty }{G}_i=M$.

The topology of $M$ has a countable basis consisting of open sets $A_j$, $j=1,2,\dots$, with compact closures $G$.

$G_1=A_1$. Suppose $G_k$ has been defined as $G_k=A_1\cup \cdots \cup A_{j_k}$. Let $j_{k+1}$ be the smallest natural number for which$${\overline{G}}_k\subset A_1\cup \cdots \cup A_{j_{k+1}}$$Define $G_{k+1}:=A_1\cup \cdots \cup A_{j_{k+1}}$. This sequence of $G_k$ has all the properties required by Step 1.

Step 2: Given the open covering of $M$ by the $U_i$, we can choose a chart $(V_x,{\phi }_x)$ for every $x\in M$, so that $x\in V_x$, ${\phi }_x(V_x)=B(y_x,3)$.

Let $W_x={\phi }_x^{-1}(B(y_x,1))$. We may assume the $V_x$ form a refinement of the $U_i$, i.e. $\forall x$, $\exists i=i(x)$, s.t. $V_x\subset U_i$.

Each set ${\overline{G}}_k\setminus G_{k-1}$ can be covered by finitely many such $W_{x_i}$, $i\in \{1,\dots ,l\}$, such that, moreover,$$V_{x_i}\subset G_{k+1}\setminus {\overline{G}}_{k-2}=G_{k+1}\cap (M\setminus {\overline{G}}_{k-2}).$$

Now let $x\in M$ $\Rightarrow$ $m\in G_i$. Take $G_i\setminus {\overline{G}}_{i-1}$. This will intersect only finitely many $V$’s. So $V_{x_i}$ are a locally finite refinement.

$\square$

例 4.1.7. Let $M$ be compact. $\forall x\in M$, $\exists$ a chart of this form around $x$, $\{W_x\mid x\in M\}$ is an open covering of $M$. Because $M$ is compact, $\exists x_1,\dots ,x_l\in M$, s.t. $\bigcup _{i=1}^l{W}_{x_i}=M$.

4.2Partition of Unity

定义 4.2.1. Let $M$ be a smooth manifold. Let $\{U_i\mid i\in I\}$ be an open covering of $M$. A smooth partition of unity on $\mathbf{M}$ subordinate to the covering $\{{\mathbf{U}}_{\mathbf{i}}\mid \mathbf{i}\in \mathbf{I}\}$ is a collection of smooth functions ${\rho }_j:M\to \mathbb{R}$ such that ${\rho }_j⩾0$, $\forall j\in J$ for which the supports of the ${\rho }_j$ form a locally finite refinement of the $U_i$ and $\sum _{j\in J}{\rho }_j\equiv 1$.

定义 4.2.2. If $f:M\to \mathbb{R}$ is any continuous function, define $\mathrm{supp}(f)=\overline{\{x\in M\mid f(x)\ne 0\}}$.

In the definition of partition of unity, we want $\mathrm{supp}({\rho }_j)\subset U_{i(j)}$.

定理 4.2.3. If $M$ is any smooth manifold and $\{U_i\mid i\in I\}$ is any open covering, then there is a subordinate smooth partition of unity.

证明. First consider the following smooth function $f:\mathbb{R}\to \mathbb{R}$.

$$f(x)=\{\begin{array}{rlr}& \exp \left(\frac{1}{x(x-1)}\right)& \text{for}0⩽x⩽1\\ & 0& \text{otherwise}\end{array}$$is a ${\mathcal{C}}^{\infty }$ function, a bump function.

$$g(x)=\frac{{\int }_{-\infty }^{x}f(t)\mathrm{d}t}{{\int }_{\mathbb{R}}f(t)\mathrm{d}t}$$is also ${\mathcal{C}}^{\infty }$.

Given the $U_i$, construct an atlas in the proof of the Proposition 4.1.6 $\{(V_k,{\phi }_k)\mid k\in K\}$.

Define ${\rho }_k:M\to \mathbb{R}$ so that it is ${\mathcal{C}}^{\infty }$ and$$\begin{array}{r}{\rho }_k{|}_{W_k}\equiv 1\\ {\rho }_k⩾0\\ \mathrm{supp}({\rho }_k)\subset V_k\end{array}$$The supports are thus a locally finite refinement of $U_i$ and $s=\sum _{k\in K}{\rho }_k$ is defined everywhere$>0$.

Define ${\overline{\rho }}_k:=\frac{{\rho }_k}{s}$, $\sum _{k\in K}{\overline{\rho }}_k\equiv 1$.

$\square$