Connected Set in $ {R}^{2} $ Is $A=\{(x,y): x^2+y^2=1\}$ 
is  connected in $ℝ^2$? 
From its graph, I would conclude that it's not path connected. 

$$
A = \left\{(x,y) : x^2 + y^2 = 1 \right\} = \left\{(x,y) : x = \cos \theta, y =  \sin \theta, 0 \leq \theta < 2\pi\right\}
$$
The interval $\Theta = \left[0,2\pi \right)$ is connected, since $A$ is image of continuous map defined on a connected set then it is also connected (as proved here)
