Understanding the Difference Between Open Balls and Neighborhoods

  • Context: Graduate 
  • Thread starter Thread starter mynameisfunk
  • Start date Start date
  • Tags Tags
    Balls
Click For Summary
SUMMARY

Open balls and neighborhoods are not identical concepts in topology. An open ball is defined specifically within a metric space, while a neighborhood refers to an open set containing a point. The term "neighborhood" implies a focus on smaller sets, contrasting with the broader notion of open sets. There are three distinct definitions of a neighborhood of a point x: an open ball around x, an open set containing x, and a set with an open subset that contains x.

PREREQUISITES
  • Understanding of metric spaces
  • Familiarity with open sets in topology
  • Basic knowledge of epsilon-delta definitions
  • Concept of subsets in set theory
NEXT STEPS
  • Study the properties of metric spaces in detail
  • Learn about open sets and their significance in topology
  • Explore epsilon-delta definitions in calculus
  • Investigate the concept of convergence in metric spaces
USEFUL FOR

Mathematicians, students of topology, and anyone interested in the foundational concepts of metric spaces and open sets.

mynameisfunk
Messages
122
Reaction score
0
Are open balls and neighborhoods the exact same thing? If not, could you please shed some light on this for me?
 
Physics news on Phys.org
A neighborhood is just an open set. An open ball requires being in a metric space. The word neighborhood is usually used as opposed to just open set because you want to give the impression that the open set is supposed to be a small one, similar to saying let [tex]\epsilon>0[/tex] vs saying let [tex]M>0[/tex]. They both say the exact same thing but one of them indicates we're interested in picking small numbers and one large numbers. It's not a formal definition but just to give the reader some intuition
 
There are at least three inequivalent definitions of "neigborhood of x":

1. An open ball around x.
2. An open set that contains x.
3. A set that has an open subset that contains x.
 

Similar threads

High School Roulette wheel physics and probability
  • · Replies 9 ·
Replies
9
Views
7K
MHB Carothers' Definitions: Neighborhoods, Open Sets, and Open Balls
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Ways to put n balls in m boxes such that exactly k boxes are empty?
  • · Replies 29 ·
Replies
29
Views
6K
Undergrad Open balls dense in closed balls in Euclidean space
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Why is [0,1] compact in the case of a collection of open neighborhoods?
  • · Replies 9 ·
Replies
9
Views
6K
MHB Solve Probability Problem: An Urn with 5 Black & 4 White Balls
  • · Replies 3 ·
Replies
3
Views
2K
High School How to characterize a rubber ball moving horizontally and bouncing off of a vertical wall?
  • · Replies 8 ·
Replies
8
Views
2K
Undergrad How Many Ways Can You Distribute 4 Indistinguishable Balls into 4 Boxes?
  • · Replies 22 ·
Replies
22
Views
4K
Undergrad What is the correct expectation value for this game with redraw?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Can this experiment break Lorentz symmetry?
  • · Replies 35 ·
2
Replies
35
Views
2K