Interior of a connected set is connected

Question

I know interior of connected set in a metric space need not be connected. Simplest example would be to take two tangent closed disk in Euclidean plane.

I am trying to construct a counterexample in $\mathbb{R}$? I mean I am trying to find a connected set in $\mathbb{R}$ such that interior is not connected.

Thanks for reading, and helping out.

Answer 1

In $\mathbb{R}$ a set is connected if and only if it is path connected. It follows that the only connected subsets are the intervals $I\subseteq \mathbb{R}$. The interior of a closed interval $[a,b]$ is $(a,b)$, which is again connected. The other cases are the same.

So, every connected subset of $\mathbb{R}$ has connected interior.