Differential Geometry | Topology Answers

Problem:

Part A: Let X be a topological space and ∞ / ∈ X. Define b X = X ∪{∞} and consider on b X the family: T = {A ⊂ X| A is open in X}[nb X \K| K is closed and compact in Xo. 1. Show that (b X,T) is a compact topological space. It is henceforth called a one-point compactification of X. 2. Show that b X is Hausdorff if and only if X is Hausdorff and every point in X has a compact neighborhood. 3. Show that a compact Hausdorff space Y coincides with a one-point compactification of Y \{y} ∀y ∈ Y . 4. Show that if f : X → Y is a continuous map such that the pre-image f−1(K) of any compact K ⊂ Y is a compact in X, f extends naturally to a continuous mapb f : b X →b Y .

Let (X,T) be a Hausdorff space and K = {A ⊂ X| A∩K is open ∀K ⊂ X compact}. Show that K is a finer topology than K, and that (X,K) is compact.

Let X and Y be two topological spaces with Y Hausdorff, and let f,g : X → Y be two continuous maps. Show that: (a) If f is one-to-one, then X is Hausdorff; (b) If f(x) = g(x) ∀x ∈ A, where ¯A = X, then f = g; (c) The graph Γ = { (x,f(x)) | x ∈ X} is closed in X ×Y .

A particle moves so that its position vector at time t is given by r =e−t(costi +sintj). Show that at any time t, (a) its velocity v is inclined to the vector r at a constant angle 3 (b) its acceleration vector is at right angles to the vector r.

Show that the gradient vector is perpendicular to the tangent at a point of an isovalue curve.

Solve the following initial value problem by using Laplace transform 2d^2y/dt^2 +3 dy/dt - 2y=te^-2t, y(0)=0, y’(0)=-2

Show that any open disc in xy-plane is a surface

A curve is uniquely determined except as to position in space when its curvature and torsion are given functions of arc length S?