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 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? (Please see attachment for full question)

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 . a) Show that, is open then is open. b) Show that f is an identification map. or, Let A: {xEn : x≥0} and let f:n → A be continuous map given by f(x):=|x|... See attachment for full question.

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 group problem

Problem: 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. Need a detailed proof outline and a summary of background information necessary for the proof