Intersection of path-connected sets in $\mathbb{R}^{n}$

Question

Let $A, B \subseteq \mathbb{R}^{n}$ be two path-connected sets. Is it true that $A \cap B$ is also path-connected?

(A subset of $\mathbb{R}^{n}$ is path-connected if every pair of points in $A$ can be joined by a path in $A$).

Intuitively, I think that the answer is no. I don't know how to disprove this though. Can someone please help me with this exercise? I am having trouble finding a counterexample.

Answer 1

Take the two (closed) halves of a circle. Each of them is path-connected, but their intersection isn't even connected.

Answer 2

Visual proof of the answer by Jose' Carlos Santos: