Exercise 4.22

Question

Exercise 22:
    Let $A$ and $B$ be disjoint nonempty closed sets in a metric space $X$ and define $$f(p)=\frac{\rho_A(p)}{\rho_A(p)+\rho_B(p)}\quad(p\in X).$$ Show that $f$ is a continuous
    function on $X$ whose range lies in $[0,1]$, that $f(p)=0$ precisely on $A$ and $f(p)=1$ precisely on $B$. This establishes a converse of Exercise 3: Every closed set
    $A\subset X$ is $Z(f)$ for some continuous real $f$ on $X$. Setting $$V=f^{-1}\bigl([0,1/2)\bigr),\quad W=f^{-1}\bigl((1/2,1]\bigr),$$ show that $V$ and $W$ are open and
    disjoint, and that $A\subset V$, $B\subset W$.

analambanomenos · Accepted Answer

If $\rho_A(p)+\rho_B(p)=0$, then $\rho_A(p)=\rho_B(p)=0$. By Exercise 20(a) this implies that $p\in\bar A\cap\bar B=A\cap B$. But $A$ and $B$ are disjoint, hence $\rho_A(p)+
    \rho_B(p)>0$. Hence, since $\rho_A$ and $\rho_B$ are continuous on $X$, $f(p)$ is also continuous on $X$ by Theorem 4.9. Also, $f(p)=0$ if and only if $\rho_A(p)=0$, that is,
    $p\in\bar A=A$, and $f(p)=1$ if and only if $\rho_B(p)=0$, that is, $p\in\bar B=B$.

Since $[0,1/2)$ and $(1/2,1]$ are disjoint open sets in $[0,1]$ and $f$ is continuous on $X$, $V$ and $W$ are disjoint open sets of $X$ by Theorem 4.8.

Exercise 4.22

Answers

Comments

Exercise 4.22

Answers

Comments

Add answer