TOPOLOGY - KING OF MATHEMATICS

                 TOPOLOGY - KING OF MATHEMATICS

          From our childhood we are studying geometry, there we are studying different shapes and objects and their properties like area and many. Although in mathematics we are not bounded by dimension so we make any dimensional object and study it. By studying this in higher dimension we know more about real world and we more understand more pattern, this patterns are always in front of us but we have to find this.

         Although we are saying in any dimension we are able work in any objects is it not that if we choose any arbitrary shape its sometimes difficult to study this formally. 

        Many days earlier I am familiar with another branch of mathematics there we don't have this boundary. Although by doing these process we have to make our geometry deform or properly say we not that much focus on geometry.

Wait ! What I mean by that ?

When we study geometry of any shape or any geometric object what is our basic tool to study all this???

 Any answer …..

Some of u may say numbers (Yeah ! you right but I ask you think more deeply what this numbers are exactly ??)

If u deeply think this numbers are nothing but distance between two points. 

Like example In circle what is radius?

It is nothing but distance between centre of a circle and any point on a circumference.

Think in triangle what its sides are nothing but distance two points if we join the three points by a line which is distance we gave its form a tringle.

So distance is the basic tools for every geometric study.

Now u can define distance between points by many ways.

Mathematicians called this study as a study of Metric Space. Where we deal with our usual arbitrary space with given different distance.

Example - we all are familiar with Real Analysis its a kind of metric space where my space is Real Number and distance between any point is same as our normal thinking , we mathematician called it usual metric or Euclidian metric on R. 

Now u may wonder how distance can different between two point if I always took the same space ???

So there should be proper defining of this field that makes the study more formal, we mathematicians called this metric space property, that are following ..

We mathematicians formally define it as a function , d : X * X --> R by (where X is my chosen space and R is the set of real number)

1) d(x , y) ≥ 0 and d(x , y) =0 if and only if x = y , ∀  x , y ∈ X

2) d(x , y) = d(y , x), ∀  x , y ∈ X

3) d(x , y) ≤ d(x , z) + d(z , y), ∀  x , y, z ∈ X

Now come to the question how we define different distance in any same space. I think your mathematical mind gives u some answer ….

Basically we define our metric space like a way that our usual distance thinking.....

Let try some experiment define d : R * R --> R

d(x , y) =0 if x=y 

            = α if x ≠ y where α > 0

U can check its satisfy above all property or not...

If α = 1 we mathematicians called this metric space as Discrete Metric Space. 

Now just imagine what is the distance between any two different point its nothing but 1 (what !!!)

Distance between 1 and 2 is 1 and distance between 1 and 100000000000000000000000000000000000000000000000000000000000000 is also 1. What u feel is that,  isn't it wonderful. Here is the beauty of mathematics.

Just try to imagine how it looks like. 

So in above  we see that different distance can define on same space, but as I say in first although this distance gives us wonderful tool still we can't do  more arbitrary work on it in formal way. 

So that the time we sacrifice our most wonderful tool that is distance (it's say if u not sacrifice your comfort u couldn't get more) .Now rather not focusing on distance we are focus on the neighbourhood containing that point, that idea gives us a wonderful idea or more formal way to overcome our problems not only that, more different aspect of looking an arbitrary space, we mathematicians called Topology or Topological space.

As we previous define metric space formally topological space also has some property to satisfy.

I am not going to this, just end with an wonderful example to give u idea the power of the topological space.

Triangle in 3D mean all inside triangle also, u know geometrically, in metric space it has some distance between any two points of inside of triangle that distance we cannot change but in topology we are not bother about distance so we can whatever we can. Lets deform the tringle into circular shape. As we can do like take some clay make a triangle then make a circle, u noticed that we make the circle by same clay so this thing wondered me many days this a kind of study we do in topology where above two geometric shape are same or formally they are homeomorphic.