Exercise 4.21

Question

Exercise 21:
    Suppose $K$ and $F$ are disjoint sets in a metric space $X$, $K$ is compact, $F$ is closed. Prove that there exists $\delta>0$ such that $d(p,q)>\delta$ if $p\in K$, $q\in F$.
    Show that the conclusion may fail for two disjoint closed sets if neither is compact.

analambanomenos · Accepted Answer

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Since $ho_F$ is a continous function on the compact set $K$, by Theorem 4.16 it must attain its minimum value on $K$ at some $x\in K$. This minimum value $\delta$ must be
    nonzero, since if $ho_F(x)=0$ then by Exercise 20(a) $x\in\bar F=F$ which contradicts the disjointness of $K$ and $F$.

An example where this fails for two planar closed sets is 
    \begin{align*}
        F_1 &= \{(x,y)\in{\mathbf R}^2|x<0,y\ge 1/x^2\} \
        F_2 &= \{(x,y)\in{\mathbf R}^2|x>0,y\ge 1/x^2\} 
    \end{align*}
    A one-dimensional example is
    \begin{align*}
        F_1 &= \hbox{set of positive integers} \
        F_2 &= \cup_{n=0}^\infty \bigl[n+2^{-(n+2)},n+1-2^{-(n+2)}\bigr] \
        &= 	extstyle [\frac{1}{4},\frac{3}{4}]\cup[1\frac{1}{8},1\frac{7}{8}]\cup[2\frac{1}{16},2\frac{15}{16}]\cup\cdots
    \end{align*}

Exercise 4.21

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Exercise 4.21

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