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12 CHAPTER I. Manifolds and Vector Fields 1.16. Corollary. (1) The (separable) connected smooth manifolds are ex- actly the smooth retracts of connected open subsets of Rn 's. (2) A smooth mapping f : M — » N is an embedding of a submanifold if and only if there is an open neighborhood U of f(M) in N and a smooth mapping r : U — M with r o f = MM- Proof. Any manifold M may be embedded into some Mn see (1.19) below. Then there exists a tubular neighborhood of M in Rn (see later or [84, pp. 109-118]), and M is clearly a retract of such a tubular neighborhood. The converse follows from (1.15). For the second assertion we repeat the argument for T V instead of Mn. • 1.17. Sets of Lebesque measure 0 in manifolds. An m-cube of width w 0 in E m is a set of the form C = [#i, x\ + w] x ... x [xm, xm + w\. The measure /JL(C) is then /i(C) = wn. A subset S C Rm is called a set of (Lebesque) measure 0 if for each e 0 these are at most countably many m-cubes d with S C I J £ o ^ anc ^ X^SoM^) £ - Obviously, a countable union of sets of Lebesque measure 0 is again of measure 0. Lemma. Let U C Rm be open and let f : U - Rm be C1. If S C U is of measure 07 then also f(S) C Mm is of measure 0. Proof. Every point of S belongs to an open ball B C U such that the operator norm ||d/(x)|| KB for all x G B. Then \f(x) — f(y)\ KB\X — y\ for all x, y G B. So if C C B is an m-cube of width w, then /(C) is contained in an m-cube C of width ^s/rnKBW and measure Now let S = [JjLi Sj where each Sj is a compact subset of a ball Bj as above. It suffices to show that each f(Sj) is of measure 0. For each e 0 there are m-cubes Ci in Bj with Sj C (Ji @i a n d Yli M(Ci) £: - As we saw above, then /(X^) C (Ji C^ with ^ - /i(^) mml2K^.e. D Let M be a smooth (separable) manifold. A subset 5 C M is called a se£ o/ (Lebesque) measure 0 if for each chart ({7, u) of M the set u(S fl J7) is of measure 0 in E m . By the lemma it suffices that there is some atlas whose charts have this property. Obviously, a countable union of sets of measure 0 in a manifold is again of measure 0. An m-cube is not of measure 0. Thus a subset of W71 of measure 0 does not contain any m-cube hence its interior is empty Thus a closed set of measure 0 in a manifold is nowhere dense. More generally, let S be a subset of a manifold which is of measure 0 and cr-compact, i.e., a countable union of compact subsets. Then each of the latter is nowhere dense, so 5 is nowhere dense by the Baire category theorem. The complement of S is residual, i.e., it contains the intersection of a countable family of open dense subsets.