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However, I want to know if this is another example:

The covering space C is a direct sum (tagged union) of S^1 and S^1. So, each point in C is just a point on one of two disjoint circles. C is disconnected, obviously.

The covering map from C -> S^1, then, would just be the map which throws away the information about which circle the point came from. Graphically, it would be similar to the spiral idea, except instead of a spiral casting shadows, it's one circle above another.

As far as I can tell from the definition, this set C is a covering set, but since there are few examples in my book, I just wanted to confirm. I keep feeling like C being disconnected might be a problem, since all the examples I've seen in my book and elsewhere all have the covering space and the topological space in question with an equal number of disconnected parts.