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Really stumped on this problem. How should I go about showing that the earrings are connected? Just claim that a circle is connected?

To show it is path connected, we need to cook up a function that maps an interval to the circle, right?

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Your path from one point $x$ to another can go around the circle you start on to the common point, and then along the circle of the other point $y$ of the earring space. –  coffeemath Nov 14 '12 at 4:26
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And note that path-connectedness implies connectedness. –  Brian M. Scott Nov 14 '12 at 4:36
    
to prove path connectedness what function would you use. Motivate the proof for me, if you can Brian. For some reason, I am having a lot of trouble visualizing it. –  user43901 Nov 15 '12 at 1:18

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