Let A = {(0,y) in R^2 : y in R} Let B = {(x,y) in R^2 : (x-1)^2 + y^2 < 1} Is A connected? Is B connected? Is AUB connected?
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The set A is simply the y-axis. The set B is the interior of a disk centered at (1,0) with radius 1. Take note that (0,0) is a limit point of B as well as a point of A.
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