MATH 381 Midterm: MATH 381 McGill Examw97

• School: McGill University
• Department: Mathematics and Statistics
• Course: MATH 381
• Professor: All

Document Summary

Mathematics 189-354a: carefully state the following theorems. (i) (5 marks) the heine borel theorem. (ii) (5 marks) the baire category theorem. (iii) (5 marks) the tietze extension theorem. (iv) (5 marks) the picard existence theorem. Show that there exists a strictly positive real number such that for each x x, there exists i such that u(x, ) u . Here the notation u(x, ) stands for { ; x, d(x, ) < }. 4. (i) (7 marks) let a and b be disjoint closed subsets of a metric space x. Show that there necessarily exist disjoint open subsets u and v such that a u and. Show that inf{ka bk; a a, b b} > 0. Here k k denotes the euclidean norm on rd. 5. (i) (10 marks) let be a connected open subset of rd. (x, y) = (x y2)(x 3y2) does not have a strict local minimum at the origin (x, y) = (0, 0).