Give an example of a set A and a point $x \notin A$ such that d(x,A)=0

Question

This question is very vague, but I am confused about the metric $d$. A metric on a set $X$ is defined as $d: X \times X \to [0, \infty)$ but can we define a metric such that $d: X \times X^C \to [0, \infty)$? by the way, the solution says $x=0$ and $A = (0, \infty)$.

Answer 1

When $X$ is a metric space with metric $d$, $x\in X$ and $A$ is a nonempty subset of $X$, then $d(x,A)=\inf\{d(x,y):y\in A\}$ by definition.

You should be able to check that $d(x,A)=0$ for $x=0$ and $A=(0,\infty)$ inside the reals with the usual metric.

Answer 2

Yes, that solution goes through because $x_{n}=1/n\in(0,\infty)$ are such that $d(0,A)=d(0,(0,\infty))\leq d(0,x_{n})=d(0,1/n)=1/n\rightarrow 0$, so $d(0,A)=0$.