1 / 8

Topology

Topology. Uniformly Continuous. The field of topology is about the geometrical study of continuity. A map f from a metric space ( X , d ) to a metric space ( Y , D ) is continuous if: Given e > 0,  d > 0, If d ( x 1 , x 2 ) < d , then d (f( x 1 ), f( x 2 )) < e . Sequences.

buck
Download Presentation

Topology

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Topology

  2. Uniformly Continuous • The field of topology is about the geometrical study of continuity. • A map f from a metric space (X, d) to a metric space (Y, D) is continuous if: • Given e > 0, d > 0, • If d(x1, x2) < d, then d(f(x1), f(x2)) < e.

  3. Sequences • Map from positive integers to a set; s: Z+→X • nth point is xn s(n) • Sequence {xn} converges to y • given e > 0 there exists m • d(xn, y) < e for nm • Limit point p in YX • There exists a sequence of points in Y converging to p X yn y1 p Y

  4. A set YX is closed if it contains all its limit points. The closure of Y, cl(Y) is the set of all limit points in Y. Ball of radius r centered at x B(x, r) = {y: yX, d(x, y) < r} A set UX is open For each xUr The ball B(x, r)  U Radius r may depend on x Open and Closed Closed set includes its boundary Open set missing its boundary

  5. The interior of SY Int(S) = Y – cl(Y - S) The neighborhood of a point xX NX x int(N) Neighborhood Y Int(S) N x X

  6. A function f is continuous if X and Y are metric spaces The function f: XY The function f-1: YX  open VY, f-1(V) is open A homeomorphism f If f is continuous and f is invertible and f-1 is continuous Homeomorphism X f f-1 V Y

  7. Scalar field maps from a space to the real numbers. F = f(x1, x2, x3) A constraint can reduce the variables to a surface. F(x1, x2, x3) = 0 F = f(u1, u2) Smooth fields are measured by their differentiability. Cn-smooth is n times differentiable Smoothness F is not C2smooth is C2 smooth

  8. A function f is class Cn XEa and YEb The function f: XY Open sets UX, VY The function F: UV Partial derivatives of F are continuous to order n F and f agree on X A function fis smooth Class Cn for all n fis a Cn-diffeomorphism and f is invertible and f-1 is class Cn Diffeomorphism U F X f Y V next

More Related