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**Additional resources for Algebraic Topology: Oaxtepec 1991**

**Example text**

42 I. 3. For each integer Y > 0 let Q be divided by a network of congruent nonoverlapping n-cubes with edges of length l/r. 2. Let z,* be any one of the cubes 2; which contains a point u E Q. , A,) such that u is a critical point of the dot product A * F. Let S, be the system of coordinate xi-axes. ,yn). 25), I y, I p d p ) . Recall that the mapping t --+ ~ ( tdepends ) only on the values of F in an open convex neighborhood of Q and not upon the choice of u E u. , n - 1, in S,, of the point F(x) is similarly obtained by projecting the vector F(x) - F(u) orthogonally onto the y,-axis.

40 ANALYSIS OF NONDEGENERATE FUNCTIONS are concerned with an open subset X of En and a closed n-cube Q included in X. There is given a real-valued function f of class Ca defined on X. The Hessian H(x) off at a point x E X is the Jacobian at x of grad f. , a,) in a Euclidean space A, “gradient lemma” follows. 0. 3) of A, has the Jordan content 2ero in A , . 1. 4) of A, has the Lebesgue measure zero in En . 2. 5) is ND. 0. 0 was confirmed around 1930 in another way. 1, p. 243, on focal points of extremals transverse to a manifold.

However, the differentiable manifolds most relevant to the developments of Part I are differentiable manifolds in E, which are “regular” in E, . We shall define these manifolds in a manner which will show that they are differentiable manifolds in the sense of the general definition of Part 11. Our definition of regular manifolds will show that two of the conditions which must be imposed on the general manifolds of Part I1 are automatically satisfied by regular manifolds in E, . We refer to the “compatibility” and the “countability” conditions.