devotion & mathematics

                    In the previous section we see the convergent and Cauchy sequences in a metric space with their properties. We see that a limit point is a point to which a sequence is convergent. Also every convergent sequence in a metric space is a Cauchy sequence and  the converse of the theorem i.e., every Cauchy sequence is not convergent.We illustrate it  with an example by considering (0,1] as metric space and a sequence {1/n} which is Cauchy but not convergent in (0,1] in detail. In the following there is an introduction to few points and few definitions. Now take a look upon them. Contents : 1. Limit vs limit point 2. Every where dense or dense set 3. Nowhere dense set 4. Boundary and boundary point of a set 1 . Limit vs limit point :-       Some times we often use the terms limit and limit points. Are they both have same meaning? This is a confusion to every reader.  On the real line...

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...