Neighborhood in Mathematics

Hey there! This post talks to you about the term ‘Neighborhood in mathematics’ (nbd). which plays key role in every math stream. To know about neighborhood, we have to know limit& basic things such as limit point, interior point. also, here we’ll discuss about its types- Circular nbd & Rectangular nbd..

We’ll go step wise as further-

  • Introduction
  • Necessity
  • Definition
  • Types

Neighborhood-Introduction:

In most of the areas of mathematics, such as- Algebra,Topology,etc neighborhood is the basic & most important term. without which no function can be defined. This term resembles with the general meaning of ‘Neighborhood’ i.e. nearby things. Here, in mathematics, instead of things, we mention some points nearby a point.

In this definition, we can restrict the distance from an arbitrary point under consideration, so that the neighborhood must have some definiteness, about which we’ll discuss further.

Necessity of Neighborhood:

The importance of this term is highlighted in graphical representation. & the actual existence of the properties related to that arbitrary point (or set,say) under consideration can be clarified using nbd. Also, some major definitions such as-Limit,Cauchy sequence,etc make use of it.

Definition:

Let x0 be a fixed point & δ be a distance from x0 which is a positive real number.Then the set of all points which are at a distance δ from x0, is the definition of Neighborhood at distance δ.i.e.δ-nbd (Nδ)

In the form of set-

Nδ={x/|x-x0|<δ}

Neighborhood-def
Definition of neighborhood

Definition in Complex plane:

Any open circle around a point z0 at a distance R is its r-nbd. i.e.-{z/|z-z0|<R}

Definition in Metric space:

A neighborhood of a point ‘p’ in a metric space X, is a set Nr(p) i.e.r-nbd of p. We define it as-Nr(p)={q/|d(p,q)<r} ; r>0.

Where, r is the radius of the neighborhood & Nr(p) is the r-nbd.

Types:

There are two main types of neighborhood:

  • δ-nbd
    • Circular nbd
    • Rectangular nbd
  • Deleted δ-nbd

δ-neighborhood:

Let x0 be a fixed point & δ be a distance from x0 which is a positive real number.Then the set of all points which are at a distance δ from x0, is the definition of Neighborhood at distance δ.i.e.δ-nbd (Nδ)(neighborhood).
In the form of set-

Nδ={x/|x-x0|<0}

 

Circular neighborhood:

Set of all points P(x,y) which are at a distance δ from P(x0,y0) so as to form shape of a circle. i.e.-

Nδ={(x,y)∈R2\(x-x0)2+(y-y0)2<δ}

Circular neighborhood

Rectangular neighborhood:

Set of all points P(x,y) within a rectangle of sides δ1 & δ2 from P(x0,y0). that is-

Nδ={(x,y)∈R2\(x-x0)21,(y-y0)22}

Rectangular neighborhood

Deleted δ-neighborhood:

When |x-x0|>0, also, for x≠x0, in other words, x0 is neglected from nbd. the set so formed by combining |x-x0|>0 and  |x-x0|<δ. i.e.-

0< |x-x0|<δ that means, x0-δ , x0


This is all about the topic… Get the knowledge of limit & continuity in our recent posts.. We will discuss about more topics in next posts..