devotion & mathematics

https://youtu.be/-Z7sRAcTBRs?si=ezkfp_zlrrINstER  RESULTS ON OPEN SET:                                                                             Result :   A subset G of a metric space X is open iff it is a union of open spheres. proof :                               R   Result .    Let X be a metric space. Then i.    Any union of open sets in X is open in X and ii.   Any finite intersection of open sets in X is open in X. Result : In any metric space X, every closed sphere is a closed set. #results #open #& #closed #sets People also search for Iit mathematics  books IIT Mathematics  Questions Iit mathematics  syllabus Arihant  IIT  JEE m...

   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....

  OPEN & CLOSED SETS In this session we see open & closed sets with their properties. OPEN SET : -      A non empty set A in the real line R is called an open set , every point in A is an interior point of A.                                                                     Examples:- * (a, b)  is an open set.  * [a, b] is not open since the end points a, b are not interior points. * Every finite set is not open. * The sets N, Z and Q are not open sets. * The set R of all real numbers is open. * The set of complex numbers |z|<1 is open. * The set of complex numbers |z|  ≤1 is not open. * Every neighbourhood is an open set. Properties of open sets:- 1. Every open set is a union of open intervals. 2. Union of two open sets is open. 3. Intersection of two open sets is ...