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Define the obvious quotient map p : 0,x R" + u,, which associates to each element its equivalence class. 5 of the equivalence relation). In other words, in the diagram where (1 means the identity) no pair of distinct points has the same image under p, so that the quotient map i is well defined, with i p = 1, 0 It follows immediately that p and i are inverses of one another, so that U , is homeomorphic to R" x R" = R'". It follows, without difficulty, that TM" is Hausdorff and has a countable base.

The map n:S" --* RP", which associates to each point its equivalence class, consisting of the point and its antipode, is onto. Clearly if y # x and y # -x, then n ( y )# n(x). It follows that any subset of S" that does not contain a pair of antipodal points is mapped in 1-1 fashion by n. In particular, let U , be the set of points y E S", so that the angle, at the origin 0, of the segments from 0 to x and from 0 to y is less than n/2; it is mapped by II, in a 1-1 fashion, onto a subset of RP".

The algebra of differentiable functions in a local setting is the theory of germs and jets. This theory, which we now look at briefly, has become quite fashionable; it is, in fact, rather more than merely convenient language. GERMS AND JETS If one wishes to study the local behavior of functions near a point in Euclidean space, one might naively look at the function at a single point. This would amount to a forfeiture of all information about derivatives; to understand derivatives one has to look at a function in an open neighborhood of a point.