Exercise 10.5

We want to show: Suppose $K$is a compact subset of a metric space $X$, and $\{V_{\alpha }\}$is an open cover of $K$. Then there exists ${\psi }_1,\dots ,{\psi }_s\in C(X)$such that

• $0\le {\psi }_i\le 1$for $1\le i\le s$;
• each ${\psi }_i$has its support in some $V_{\alpha }$, and
• ${\psi }_i(x)+\cdots +{\psi }_s(x)=1$for every $x\in K$.

Repeating the proof of Theorem 10.8 in the text and following the hint, associate with each $x\in K$ an index $\alpha (x)$ so that $x\in V_{\alpha (x)}$. Then there are open balls $B(x)$ and $W(x)$ centered at $x$, with

Since $K$ is compact, there are points $x_1,\dots ,x_s$ in $K$ such that

By Exercise 4.22, there are functions ${\phi }_1\dots ,{\phi }_s\in C(X)$ such that ${\phi }_i(x)=1$ on $\overline{B(x_i)}$, ${\phi }_i(x)=0$ outside $W(x_i)$, and $0\le {\phi }_i(x)\le 1$ on $X$, namely,

where ${\rho }_{i1}(x)$ is the distance from $x$ to the complement of $W(x_i)$, a closed set, and ${\rho }_{i2}(x)$ is the distance from $x$ to $\overline{B(x_i)}$. Letting ${\psi }_1={\phi }_1$, and

for $i=1,\dots ,s-1$, the remainder of the proof follows exactly as in the proof of Theorem 10.8.