Let xEn/n be an irrational number. For each rational number yEn define... Let be the collection of all these open balls of n. Is α an open cover for n and/or n ? Does there exist a finite subcover of α for n and/or n?

Real Analysis Let xEn/n be an irrational number. For each ...continues

Let and let be continuous map given by . a) Show that, is open then is open. b) Show that f is an identification map.

Real Analysis Let and let be continuous map given by . ...continues

Metric Space

Show that (n³, d∞) is a complete metric space. d∞ is the distance in the metric space/open ball. Please see attachment.

1. Let α:={...}. Is this an open cover for (0,2)? Can you find a subcover for α for (0,2)? 2. Find an open cover for (0,2) which consists of open balls of the form B1(z) with radius 1, which does not contain any finite subcover.

1. Let α:={...}. Is this an open cover for (0,2)? Can you find a subcover for α for (0,2)? 2. Find an open cover for (0,2) which consists of open balls of the form B1(z) with radius 1, which does not contain any finite subcover. Please see attached file for full problem description.

Fundamental Groups, Path-Connected Space, Connectivity and Homotopy

Let X be a path-connected space and suppose that every map f: S^1 --> X is homotopically trivial but not necessarily by a homotopy leaving the base point x_0 fixed. Show that pi_1(X,x_0) = 0.

The Exact Homology Sequence (Exact Sequence of Triples)

Problem: Let X = X_1 / X_2, and A = X_1 / X_2. Using the exact sequence of triples, show that if the inclusion (X_1, A) --> (X, X_2) induces an isomorphism on homology, then the same holds for the inclusion (X_2, A) --> (X, X_1). Notation: X_1 is X subscript 1 / is union / is intersection --> is an inclusion map ...continues

Vectors in spherical and cylindrical

(a) Given A = a*p_hat + b*psi_hat + c*z_hat (cylindrical unit vectors), where a, b, and c are constants. Is A a constant vector (uniform vector field)? If not, find: the divergence and curl of A (b) If A = a*r_hat + b*theta_hat + c*phi_hat in spherical coordinates, with constant coefficients. Is A a constant vector (unifor ...continues

Equivalent Paths

Let f,g ; I-->X be two paths with initial point x0 and terminal point x1. Prove that f g iff f g-bar is equivalent to the constant path at x0 . Note: the path g-bar is obtained by traversing the path g in the opposite direction. ---

Isomorphism of fundamental groups

(See attached file for full problem description with proper symbols) --- a) Under what conditions will two path classes, and , from to , give rise to the same isomorphism of onto ? b) Let be an arcwise-connected space. Under what conditions is the following true: For any two points , all path classes from ...continues

Lebesque Number and Connectivity

Lemma. Let {Ui} be an open covering of the space X having the following properties: (a) There exists a point x0 such that x0 Ui for all i. (b) Each Ui is simply connected. (c) If i≠j, then Ui Uj is arcwise connected. Then X is simply connected. Prove the lemma using the following approach: To pro ...continues