What is Metric Space,Bounded Sets,Diameter,Open ball,Close Ball,Sphere Ball ?

What is Metric Space,Bounded Sets,Diameter,Open ball,Close Ball,Sphere Ball ?

You can free read a best article about  Metric Space,Bounded Sets,Diameter,Open ball,Close Ball,Sphere Ball by alamdar in pdf format.{{

 Metric Space

Let X be a non-empty set and R denote the set of real numbers.A function d:X x X⇾ R is said be a metric (or a distance function).
If it has following properties.
(M 1): For any pair x,y in X d(x,y)≥0 and d(x,y)=0 if and only if x=y.
(M 2):  For any pair x,y in X d(x,y)=d(y,x).
(M 3): For any pair x,y,z in X
d(x,z)≤d(x,y) +d(y,z).
The pair is called)  (X,d) is called then a Metric Space.

Bounded Sets

Let A and B be subsets of a metric space (X,d). The distance between A and B is defined by:

d(A,B)=inf d(a,b)

              a∈A
              b∈B
If A={x} is a singleton subset of X,then d(A,B) is written as d(x,B) and is called the distance of the point x from the set B.Thus
   d(A,B)=inf d(x,B)
                 x∈A

Open Ball

Let xo be a point in a metric space  (X,d). For any real number r > 0, the set 

 B(xo,r)={x ∈ X:d(x, xo < r}

 is called a Open Ball with Center at xo  and radius r.

Close Ball

Let xo be a point in a metric space  (X,d). For any real number r > 0, the set 

 bar of B(xo,r)={x ∈ X:d(x, xo ≤ r}

Is called  Close Ball with Center at xo  and radius r.

Sphere Ball

Let xo be a point in a metric space  (X,d). For any real number r > 0, the set 

 S(xo,r)={x ∈ X:d(x, xo = r}

is called Sphere Ball  with Center at xo  and radius r.