locally finite subset $A$ of a metric space

Question

Let $\mathcal{A}$ be a locally finite family in a metric space $ (Y, d) $.

Show that the function

$y \rightarrow$ Sup $\{d (y, Y \setminus A): A \in \mathcal{A}\}$ defined from $Y $ in $\overline{R}$ is continuous. Here $\overline{R}$ denotes the line extended.

Answer 1

HINT: Let $f:Y\to\overline{\Bbb R}:y\mapsto\sup\{d(y,Y\setminus A:A\in\mathscr{A}\}$.

Note that for any $y\in Y$ and $S\subseteq Y$, $d(y,S)=0$ iff $y\in\operatorname{cl}S$. Thus, for any $A\in\mathscr{A}$ we have $d(y,Y\setminus A)>0$ iff $y\notin\operatorname{cl}(Y\setminus A)$, which is the case iff $y\in Y\setminus\operatorname{cl}(Y\setminus A)=\operatorname{int}A$. For each $y\in Y$ let $$\mathscr{A}(y)=\{A:A\in\mathscr{A}\text{ and }y\in\operatorname{int}A\}\,.$$

• Show that $f(y)=\max\{d(y,Y\setminus A):A\in\mathscr{A}(y)\}$.
• Show that $\bigcap\{\operatorname{int}A:A\in\mathscr{A}(y)\}$ is an open nbhd of $y$ and therefore contains some open ball $B(y,r_y)$ centred at $y$.
• Show further that we can choose $r_y$ small enough so that for each $A\in\mathscr{A}$, $B(y,r_y)\cap A\ne\varnothing$ iff $y\in\operatorname{cl}A$.
• Show that if $z\in B(y,r_y)$, then $|f(z)-f(y)|\le d(z,y)$.

Use these results to show that $f$ is continuous at each point $y\in Y$ and therefore is continuous.