Mapping on induced topology and distance metric

Question

Let $(X, d)$ be a metric space. Let $τ$ be the metric topology on $X$ induced by $d$. For $A ⊆ X$ , let $d(x, A) := \inf_{a∈A} d(x, a) $ for $x ∈ X$

(a) If $f (x) := d(x, A)$ (for a fixed subset A) then show that $f : (X, τ ) → R $ is a continuous mapping.

(b) If every infinite bounded set in $X$ has a limit point (in $X$) and $A$ is closed (in $(X, τ )$ ) then show that for each $x ∈ X$ there exists an element $a_0 ∈ A $ such $d(x, A) = d(x, a_0)$.

I have no clue how should I even approach to this. A detailed solution would be great.

Answer 1

The part (a) has been asked and answered several times on this site, for example:

• Continuity of the function $x\mapsto d(x,A)$ on a metric space
• Show that the function $f: X \to \Bbb R$ given by $f(x) = d(x, A)$ is a continuous function.

For the part (b), you can try proof by contradiction. Assume that $d(x,a)>d(x,A)$ for each $a\in A$. By the definition of infimum we have that for each positive integer $n$ there exists $a_n$ such that $$d(x,a_n)<d(x,A)+\frac1n.$$ Now the set $\{a_n; n\in\mathbb N\}$ is infinite and bounded. (Can you explain why it is infinite and bounded?)

Then it has at least one limit point. Let us denote a limit point of this set by $a_0$. (In fact, this limit point is a limit of a subsequence of $(a_n)_{n=1}^\infty$.) What can you say about $a_0$?