How does connectedness in $\mathbb{R}^n$ imply path-connectedness?

Question

Every open connected subset of $\mathbb{R}^n$ is path connected. How to prove?

I think I have to use every connected, locally path connected space is path connected. But how to proceed?

Answer 1

Suppose that $A$ is open and connected. If it is empty, then it is trivially path-connected. Otherwise, take $a\in A$ and let $A^\star$ be the set of those elements $b\in A$ such that there is a path $\gamma\colon[0,1]\longrightarrow A$ with $\gamma(0)=a$ and $\gamma(1)=b$. Now, prove that $A^\star$ is open, closed and non-empty. Since $A$ is connected, it will follow that $A^\star =A$ and that therefore $A$ is path-connected.

Answer 2

Combine these three facts:

• An open subset of a locally (path-)connected space is again locally (path-)connected.

• A connected locally path-connected space is path-connected.

• $\mathbb{R}^n$ is locally path-connected.