MATH 5070-001: APPLIED ANALYSIS
Fall 1997, University of Colorado at Denver
Instructor: Prof. Andrew V. Knyazev

Test 1

Please, provide complete and detailed proofs of every statement, or give a reference to the textbook if you use a statement from the book.

1. Is number rational, or irrational?
2. Consider a set of all sequences such that every element of a sequence is either a dog, a cat, or a rabbit. Determine if the set is finite, countable, or of power of continuum.
3. Consider metric spaces and equipped with the standard euclidean metric. Are the spaces isometric? (For extra credit - homeomorphic?)
4. Find all limit points and all internal points of the set A of all positive rational numbers smaller then 1, i.e.

   

   in the metric space of all positive rational numbers Q with the standard metric r(x, y) = | x-y |.

5. For extra credit.

   Let M=QxR be a metric space of ordered pairs z=(x y) with any rational x and any real y, equipped with the metric

   

   

   Consider a set A of all pairs of positive rational numbers smaller then 1, i.e.

   

   and find its closure in M.