devotion & mathematics

 HISTORICAL BACKGROUND OF METRIC SPACES:                                      Classical analysis can be described as that part of mathematics which begins with calculus and, in essentially the same spirit, develops similar subject matter much further in many directions. It is a great nation in the world of mathematics, with many  provinces,  a few of which are ordinary and partial differential equations, infinite series, and analytic functions of a complex variable. Each of these has experienced enormous growth over a long history, and each is rich enough in content to merit a lifetime of study.                                      In the course of its development, classical analysis became so complex and varied that even an expert could find his way around in it only with difficulty....

  COUNTABLE SETS :-     Consider N, the set of all +ve integers. Any set A is called countable of ∃ a bijection between A and N. Examples :-                                                        1. The set Z of all integers is countable. 2. The set N of all +ve integers is countable. 3. The set Q of all rational numbers is countable. 4. The set of elements {(1/n) / nεN} is countable. 5. The set of all even +ve integers is countable. 6. The set of all odd +ve integers is countable. Properties of countable sets :-  1. Any subset of a countable set is countable. 2. Any subset of N is countable. 3. Union of a finite number of countable sets is countable. 4. A union of countable sets is countable. Atmost countable sets :   A set which is either finite or countably infinite is called atmost countable. properties:-  1....

   https://youtu.be/7wb85EwAMRE?si=kCE0Lzdn0dLEF0mi Here in this page, I gave you proofs of some results regarding closed sets. Result 1: The sets N,W and Z are closed. Result 2: The set Q of rational numbers is not closed. Result 3: If a point p is a limit point of a nonempty set A then each neighborhood contains infinitely many points of A. Result 4: Union of arbitrary family of closed sets is not closed. Here we see their proofs. RESULT 1: The set N of natural numbers is closed. PROOF : RESULT 2: The set Q of rational numbers is not closed PROOF :  RESULT 3:  If a point p is a limit point of a nonempty set A then each                                                          neighborhood  contains infinitely many points of A. PROOF:  RESULT 4:    Union of arbitrary family of closed sets is not closed....