Neighbourhood of a Point: Understanding "Nearness"

 The concept of neighbourhood is fundamental in topology – it formalizes the intuitive idea of "points close to x."

Definition

A neighbourhood of point x in a topological space X is any set N containing an open set U that contains x.

Formally: x ∈ U ⊆ N where U is open

Think: "any set with some breathing room around x"

Simple Examples in ℝ

For point x = 3:

  • (2, 4) is a neighbourhood ✓
  • [2, 4] is a neighbourhood ✓
  • (2.9, 3.1) is a neighbourhood ✓
  • {3} is NOT a neighbourhood ✗

Open Neighbourhood

An open neighbourhood is a neighbourhood that is itself open.

Example: (2, 4) is an open neighbourhood of 3, but [2, 4] is not.

ε-Neighbourhood (Metric Spaces)

N_ε(x) = {y | d(x, y) < ε}

All points within distance ε from x.

In ℝ: N₀.₅(3) = (2.5, 3.5) In ℝ²: N₁(0,0) = open disk of radius 1

Deleted Neighbourhood

N(x)* = N \ {x} – the neighbourhood with x removed.

Used for defining limits (what happens near x, but not at x).

Key Properties

  1. If N is a neighbourhood of x and N ⊆ M, then M is also a neighbourhood
  2. Intersection of two neighbourhoods of x is a neighbourhood of x
  3. Every neighbourhood contains an open neighbourhood
  4. x belongs to every neighbourhood of x

Important Applications

Limit Point: x is a limit point of A if every deleted neighbourhood of x contains a point of A.

Interior Point: x is interior to A if A is a neighbourhood of x.

Continuity: f is continuous at x if for every neighbourhood V of f(x), there exists a neighbourhood U of x with f(U) ⊆ V.

Quick Practice

Question: Which is a neighbourhood of 2 in ℝ?

A) {2} B) (1.5, 2] C) [2, 3) D) (3, 4)

Answer: C) [2, 3)

Contains (1.9, 2.1) which is open and contains 2.

Why It Matters

Neighbourhoods provide the language for:

  • Defining limits without ε-δ
  • Characterizing continuous functions
  • Understanding convergence
  • Describing local behavior in topology

For more topology concepts explained clearly, check out my YouTube channel "Maths Mastery with Dr. Upasana P Taneja"!

Understanding nearness, one neighbourhood at a time!
Dr. Upasana P Taneja

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