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Problem 1-1.

Let X be the set of all points (x, y) in R^2 such that y = +-1, and let M be the quotinent of X by the equivalence relation generated by (x, -1) ~ (x, 1) for all x not equal to 0. Show that M is locally Euclidean and second-countable, but not Hausdorff. (This space is called “the line with two origins”.)

Solution.