14 - Connected but not Path-Connected Set in ℝ²

Connected but not Path-Connected Set in ${\mathbb{R}}^2$

There is a connected subset of ${\mathbb{R}}^2$ which is not path-connected

Proof - Counter-Example

Consider Topologist’s Sine Curve given by

$$\{(0,y):-1\le y\le 1\}\cup \left\{\left(x,\sin \left(\frac{1}{x}\right):x\in (0,1]\right)\right\}$$