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 *f*_{t} by taking *f*_{t} (*s*) to be *F* (*t, s*), and if we do this then the loops *f*_{t} “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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