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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...

Contents : 1.        Open Sphere 2.        Open sets 3.         Results on open sets           Closed sphere cl        Closed set           Results on closed sets OPEN SPHERE :        Let X be a metric space with the metric d. If x 0 is a point of x and r be a positive real number. The open sphere S r (x 0 ) with   center x 0   and radius r is the subset of X defined by S r (x 0 ) = {x Ꜫ X/d(x,x 0 ) < r}.      NOTE : 1.        An open sphere is always non-empty ,as it contains its center. 2.        S r (x 0 ) is often called the open sphere with radius r centered on x 0 . 3.        An open sphere consists all points in X which are     “ close “ to x 0 , wit...