OPEN SPHERES AND CLOSED SPHERES #open #spheres #sets #metric #spaces #closed
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...