Download Differential topology: An introduction by David B. Gauld PDF

By David B. Gauld

Delivering classroom-proven effects, Differential Topology provides an creation to indicate set topology through a
naive model of nearness area. Its remedy features a normal research of surgical procedure, laying a fantastic starting place for extra research and enormously simplifying the class of surfaces.
This self-contained therapy beneficial properties eighty-eight necessary illustrations. Its matters comprise topological areas and houses, a few complicated calculus, differentiable manifolds, orientability, submanifolds and an embedding theorem, and tangent areas. extra subject matters contain vector fields and fundamental curves, surgical procedure, class of orientable surfaces,and Whitney's embedding theorem. appropriate for complicated undergraduate classes in introductory or differential topology, this quantity additionally serves as a supplementary textual content in complex calculus and physics classes, in addition to a key resource of knowledge for college students of mechanics.

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G •• 8n - {O}. are neither open nor closed. most open sets are not closed. most closed sets are not open. and ~ and IRn are both open and closed (and they are the only such sets; cf. Corollary 5 below). ~ = X - X is always closed; since ~ THEOREM 4. Let f : X--+ Y be a function between two topological spaces. Then conditions a and b below are equivalent and conditions c. d. e. and f are equivalent. (a) f is continuous at x . (b) For every neighborhood V of f(x) in Y. there is a neighborhood U of x in X such that feU) c V • (c) f is continuous.

C We now restrict our attention to the case where the range has dimension 1. Let f : IRm _ IR be Cr (r ~ 2). A point x E IRm is a critical point of f iff all entries of the jacobian matrix Df(x) are O. The hessian of f at x is the m x m matrix Hf(x) whose (i, j) entry is a 2f/ax. , • A critical point x of f is 1 J x (non-)degenerate iff Hf(x) is (non-) singular. Definition f: IR - IR defined by f(x) = x2 + 2x + 1 has a nondegenerate critical point at x = -1; g : IR-- IR defined by g(x) = x" has a degenerate critical ·point at x· = 0; h : IR2 IR defined by h(x,y) = x 2 + 4xy + 4y2 has a degenerate critical point at (x,y) = (0,0).

We cannot draw a better picture as K2 does not embed in lR3 • In lR~ the circle of self-intersection can be eliminated very easily • .. - (] () 1 FIGURE 13 I)IJ Differential TOpology 36 EXERCISES 1. 2. 3. Prove that the cofinite topology on an infinite set is compact but not Hausdorff. Let X be a finite set. Prove that the only Hausdorff topology on X is the discrete topology. ) Determine which of the following subsets of lR or lR 2 are compact. Justify your answers, (a) Subsets of lR : 4>; lR; {x E «> I 0 :5 x:5 I}; {x E (b) 4.

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