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### III.93 Universal Covers

Let X be a TOPOLOGICAL SPACE [III.90]. A loop in X can be defined as a continuous function f from the closed interval [0, 1] to X such that f (0) = f (1). A continuous family of loops is a continuous function F from [0, 1]2 to X such that F (t, 0) = F (t, 1) for every t; the idea is that for each t we can define a loop ft by taking ft (s) to be F (t, s), and if we do this then the loops ft “vary continuously” with t. A loop f is contractible if it can be continuously shrunk to a point: more formally, there should be a continuous family of loops F (t,s) with F (0,s) = f (s) for every s and with all values of F (1, s) equal. If all loops are contractible, then X is said to be simply connected. For instance, a sphere is ...

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