In the answer to this question, it is shown that the relative complement (defined below) of a closed line-segment is an open set. Similarly, it can be shown that the relative complement of an open ball in $\mathbb{R}^n$ is a closed set. This suggests that the relative complement bears some similarities to the regular complement.
Definition 1: Given a metric space $(X,d)$, and two points $x,y\in X$, the line segment $\overline{xy}$ is the set of points between $x$ and $y$, we say a point $z$ is between $x,y$ if $d(x,y) = d(x,z)+d(z,y)$.
Definition 2:Given a metric space $(X,d)$, a subset $S\subseteq X$, and a point $x\in X$, the relative complement of $S$ with respect to $x$, denoted by $C_{x}(S)$ is the set $$\{y\in X\mid\overline{xy} \cap S=\emptyset\}$$
Clearly, $C_x(S)\subseteq S^c$, the normal complement. I have 2 basic questions about this new concept:
Is the relative complement of an arbitrary open/closed set a closed/open set?
Is there an infinite metric space with a very bright point, $x$ such that for any subset $S$ not containing $x$, $C_x(S)= S^c$ ?
(The point is called "very bright" because rays emanating from it are able to reach anywhere)
Answer to the second question The discrete metric on $\mathbb{N}$ is an example of such a space, since here we have $\overline{nm} = \{n,m\}$, so the points in $C_0(S)$ are precisely those not in $S$ (provided $0\notin S$)