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.
2026-03-25 16:39:15.1774456755
Connected Set in $ {R}^{2} $
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$$ A = \left\{(x,y) : x^2 + y^2 = 1 \right\} = \left\{(x,y) : x = \cos \theta, y = \sin \theta, 0 \leq \theta < 2\pi\right\} $$
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)