GRE multiple-choice question: Properties of a nonempty subset S of $\mathbb{R}$. The solution of the following question is letter (C), but I did not understand why it is letter (C), could anyone clarify this for me please?



thanks!
 A: C is defining the interior of $S$, which is of course open in $\mathbb{R}$.
If you want to elminate some stuff:
E is the closure of the set, and holds only if $S$ is closed.
D is a set defining the interior of the complement, may be closed, but not necessarily. Take $S=[0,1]$. The set in question is $S^c=(-\infty,0)\cup (1,\infty)$, which is open and thus its interior is itself. This set does not contain its limit points.
B is saying the complement of $S$ is open, and $S$ is therefore closed.
A means $S$ is path connected.
A: (A) is the definition of a path connected set
(B) implies $S$ is closed (it says $S^c$ is open, which means that $S$ is closed)
(C) says the interior of $S$ is an open set
(D) says the interior of $S^c$ is closed
(E) implies $S$ is closed (intersections of closed sets are closed)

It's not (A) or (B) or (E) because not all sets are path connected or closed. For instance $(0,1) \cup (2,3)$ is not closed and there's no path from $1/2$ to $5/2$ you can take without leaving the set.
Interiors are always open (because if $x$ is in an interior of a set then that means that there is an open set $U \ni x$). So (C) is always true, but (D) isn't true because most closed sets are not open so its easy to find a counterexample.
