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....