prove that there is a continuous path between any two points in a connected, open subset

Question

Let $U$ be a connected, open subset of the plane $\mathbb R^2$ . Prove that any two points $p_0$ and $p_1$ in $U$ can be joined by a path, that is, there is a continuous map $f : [0, 1] \rightarrow U$ such that $f(0) = p_0$ and $f(1) = p_1$ .

EDIT

I started by assuming U is disconnected, but there is a path between two points in different components in U. Since U is disconnected, there is a separation A, B. THe path must intersect both A and B. Let f be the path and C be the range of f, then C intersects both A and B.

Since A and B are disjoint and closed, their complements, U-A and U-B are open and cover U. Since C intersects both A and B, it must also intersect U-A and U-B. Thus C intersects bothe closed sets A and U-A and also both closed sets B and U-B.

This implies C is not connected which contradicts that U is disconnected but there is a path between two points in different components of U.Therefor U is connected and any two points in U can be joined by a path.

I would love some feedback on this proof, is it the right approach? Is it a rigorous solution?

I am struggling with creating an epsilon ball contained in U, and how that is related to connectedness. — scoobydoobydoo, Apr 16 '23 at 20:32
I suggest: assume that $U$ is disconnected but there is a path between two points in different components of $U$. Since the set is disconnected, there is a separation $A,B$. Consider the intersection of $A$ and $B$ with the range of the path. Then…? — MJD, Apr 16 '23 at 20:36
Does this answer your question? Showing an open connected subset of $\mathbb{R}^n$ is path-connected — durianice, Apr 18 '23 at 17:07
Also, Connected open subsets in $\mathbb{R}^2$ are path connected. and Connected Open Subset of Euclidean Space is Path-Connected (ProofWiki). — durianice, Apr 18 '23 at 17:08

score 1 · Answer 1 · answered Apr 16 '23 at 20:45

1

Hint: Try defining the set $$\{p\in U,~ \text{there is a path joining $p_0$ to $p$}\}$$

answered Apr 16 '23 at 20:45

Keshav

1,620

is a path the same thing as connected? – scoobydoobydoo Apr 17 '23 at 14:04
? You defined path in your question – Keshav Apr 26 '23 at 22:43

prove that there is a continuous path between any two points in a connected, open subset

1 Answers1