devotion & mathematics

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...

contents : 1. Metric Space. 2. Usual Metric 3. Discrete Metric 4. Indiscrete Metric 5. Metric on set  of real valued continuous functions 6. Metric on  R n  Metric space :- A metric on X is a real function d of ordered pairs (x, y) of elements of X which satisfies the following conditions. (i) d(x, y)≥0 (non- negativity)      and d(x, y)=0 iff x=y. (ii) d(x, y)=d(y, x) (symmetry) (iii) d(x, y) ≤ d(x, z) +d(z, y) (transitivity) Here the space X is called a metric space and it is denoted by (X, d) and the elements of X are called the points of X.     One should always keep in mind, however, that a metric space is not merely a non- empty set: it is a non-empty set together with a metric.          There are many different kinds of metric spaces, some of which play a very significant role in geometry and analysis. A few of them are * Usual metric :-        A function d on the real line R is defined by...