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...

 In this session we see the definitions and some properties of interior of a set, derived set and closure of a set. Interior of a set :-       The interior of a subset A in R is the set of all interior points of A and it is denoted by Int(A).   i.e. Int(A) = the set of all interior points of A            Properties :-  * Int(A) is the union of all open subsets of A. * Int(A) is always open. * Int(A)  ⊆ A * Int(A)=A  iff  A is open. * Int(Int(A)) = Int(A) * Int(R) = R and Int(ⲫ) = ⲫ * Int(A∩B) = Int(A) ∩ Int(B) * Int(A) is the largest open subset of A. Examples :- 1. Int(a, b) = (a, b) 2. Int[a, b] = (a, b) 3. Int(Z) =ⲫ  ; Int (N) = ⲫ   ; Int(Q) = ⲫ 4. Int(a finite set) = ⲫ Derived set :-       The set of all limit points of a non-empty set A of R is called the derived set of A and is denoted by D(A). Examples :-  * Since empty set has no limit points, D(ⲫ)=ⲫ * D( a...

In mathetics, the terms the neighborhood of a point, interior and limit points have basic role to define open and closed sets. Before going to open and closed sets , here we see the concepts of neighborhood , interior and limit point.   ** N for set of natural numbers   ** Z is for set of integers   ** Q is for set of rational numbers   ** R is for set of real numbers NEIGHBOURHOOD OF A POINT Let  x be a point of a nonempty subset A of the real line R.  The neighborhood of x with radius ϵ is denoted by Nϵ(x) and is given by Nϵ(x) = {y Ꜫ A/|x-y|<r}  The neighborhood of x is also simply written as nbd of x or nbd(x). Also the neighborhood of x, Nϵ(x) is also called as ϵ-nbd of x. Note :      1. An open interval (a,b) is a nbd of its points.      2. A closed interval [a,b] is a nbd of its  points except for end points a and b.      3. The sets N,Z,Q are not nbd's of its points.      4...