Proof or disprove: If $C\subseteq \mathbb{R}^n$ is path connected, then the interior $C^°$ is also path connected.
As I'm studying for my upcoming Analysis II Exam, I found an exercise in an old exam. I would really appreciate, if someone could help me out here, as I really don't konw how to solve this problem. Thanks in advance!
The answer by Don Thousand in the comments is a great counter example. I'd like to explain shortly how one could arrive at this (or a similar) counter example on its own.
The simplest spaces which are not path connected are spaces which are the disjoint union of two open sets. Luckily enough, the interior of a set is also an open set. So I'd try to come up with a path connected space who's interior comprises of two disjoint open balls. To keep the space path connected, you need something which connects the two balls, but has empty interior. This easily leads to Don's example, or to the similar example I had in mind - two disjoint open balls with a line segment connecting them.