Why $d(x,A)$ metric is not a set?

Question

Let $(X,d)$ be a metric space and $A \subseteq X $ I cannot comprehend why $d(x,A)$ is a real number. Isn’t it a set that consists of metrics between $x$ and elements of $A$?

Answer 1

Logically, $d(x,A)$ could be the set $\{d(x,a)\,|\,a\in A\}$. But it is usually defined as$$\inf_{a\in A}d(x,a),$$which is a real number.

Answer 2

No !not at all, d(x,A) is basically defined as the distance of the point x and the set A in the following way $$d(x,A) = Inf(d(x,y): y in A)$$

Which is a real no. Hope it works.

Answer 3

I can understand your confusion. If there are many points in $A$, then it will be necessary to select one of those points to define the distance $d(x,A)$. The most usual definition is: $$ d(x,A)=inf_{a \in A} d(x,a) $$ That identifies at a greatest lower bound for this distance. For example, let $A = (0,1)$ and $X = \mathbb{R}$ with the metric $d(x,y)=|x-y|$. Then $$ d(4, (0,1))=3. $$ There are many other distances that might be defined, but all of them would satisfy the properties of a metric: (1) $d(x,y) \ge 0$ with equality if $x=y$; (2) $d(x,y)=d(y,x)$; and (3) the triangle inequality. Still a distance is always a real number; it comes with the territory.