# Metric Space. Continuity. Compactness.

Let $(X,d)$ be a metric space; let $A$ be a nonempty subset of $X$. For each $x \in X$, we define the distance from $x$ to $A$ by the equation $$d(x,A) = \inf\{d(x,a) | a\in A\}.$$ a) Show that the function $d(x,A)$ is continuous.

b) Show that $d(x,A) = 0$ iff $x \in \overline{A}$.

c) Show that if $A$ is compact, $d(x,A) = d(x,a)$ for some $a \in A$.

d) Let $U(A,\epsilon) = \{x \in X | d(x,A) <\epsilon \}, \epsilon>0$. Show that $U(A, \epsilon)$ equals the union of open balls $B_d (a,\epsilon)$ for $a \in A$.

e) Assume $A$ is compact and suppose $A \subset U$, where $U$ is open subset of $X$. Show that $$\exists \epsilon >0 \ni [U(A, \epsilon) \subset U].$$

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–  wj32 Oct 30 '12 at 23:41
You might get a better response if you were to explain what your thoughts are so far, and which parts are giving you the greatest problems. If we know how much you can do, we are in a better position to help. –  Old John Oct 30 '12 at 23:44
Have done nr a)! need a hint or something to start with in b,c,d,e –  John Lennon Oct 31 '12 at 0:23
This is identical with artofproblemsolving.com/Forum/viewtopic.php?t=504745 –  Martin Sleziak Nov 1 '12 at 17:08