Open connected subset of $ \mathbb R^2 $is path connected

Question

Is open connected subset of $ \mathbb{R^2} $ is path connected?

Thm: Every locally path-connected connected space is path-connected. — Henno Brandsma, Nov 26 '14 at 20:25

score 0 · Answer 1 · answered Nov 26 '14 at 20:01

0

Hint: Prove any path component must be both open and closed.

answered Nov 26 '14 at 20:01

user2345215

16,422

score 0 · Answer 2 · answered Nov 26 '14 at 20:05

Hint: The answer is yes.

To prove it, call $U$ the connected, open set, and choose a point $x$ inside. Consider $U'=\{y\in U:\text{ there exists a path from $x$ to $y$}\}$. Prove that $U'$ is open and closed in $U$.

The fact that open balls are path connected will come in handy.

Open connected subset of $ \mathbb R^2 $is path connected

2 Answers2