What is the difference between the two definitions of a neighborhood?

  • Context: Graduate 
  • Thread starter Thread starter wayneckm
  • Start date Start date
  • Tags Tags
    Definition
Click For Summary

Discussion Overview

The discussion revolves around the differing definitions of a neighborhood of a point in topology, specifically contrasting the definition as an open set containing the point versus a set that contains an open set of the point. The scope includes conceptual clarification and technical explanation regarding these definitions and their implications in mathematical proofs.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Technical explanation

Main Points Raised

  • Wayne expresses confusion over two definitions of a neighborhood: one as an open set of a point x and the other as a set containing an open set of x.
  • One participant states that a neighborhood of a point x is any set that contains x in its interior.
  • Wayne questions if the second definition is the correct one, referencing Munkres's book which avoids the second version.
  • Another participant suggests that the choice of definition is a matter of convention, indicating there is no absolute right or wrong definition.
  • Wayne raises concerns about potential confusion in proofs that depend on the choice of neighborhood definition.
  • A participant acknowledges that while there may be concerns about the validity of theorems when switching definitions, analogous statements exist since a neighborhood in the first definition is also a neighborhood in the second.
  • Another participant proposes that a neighborhood can be viewed as a collection of open balls, noting that an open set can be a neighborhood itself, and that neighborhoods can be subsets of larger sets.

Areas of Agreement / Disagreement

Participants do not reach a consensus on which definition is superior, with multiple competing views on the validity and implications of each definition remaining unresolved.

Contextual Notes

There are limitations regarding the assumptions made about the definitions of neighborhoods, and the discussion highlights the dependence on conventions used in different texts. The implications of these definitions on theorems and proofs are also noted but remain unresolved.

wayneckm
Messages
66
Reaction score
0
Hello all,


Indeed I am quite confused with the definition of neighborhood of a point x since I come across two versios of it.

The first one is simply defined as open set of x while the second one is defined as a set containing an open set of x.

Apparently these two are different notions, e.g. (x-a, x+a], in first one this is not a neighborhood while the second one it is.

So which one is the right one? Thanks very much.


Wayne
 
Physics news on Phys.org
A neighborhood of a point x is any set which contains the point x in its interior.
 
So are you saying the second version is the right one?

I could read in Munkres's book that he said "we shall avoid (the second version of neighbourhood)".
 
Yes, well it is just a convention. Given a text, you just have to figure out which convention they are using. There is no right or wrong one.
 
But won't this cause confusions and troubles when one proves with the use of neighborhood? I mean somehow the theorems may depend on the particular choice of "neighborhood"?

Apparently the second convention is more general since in some sense we can "give name to more sets".
 
you are afraid that some thm statement involving one version of "nbhd" might not be true when u replace the meaning of "nbhd" with the other version?

That's true, but there is always an analogous statement, since a nbhd in version 1 is also a nbhd in version 2 and every nbdh of version 2 contains a nbhd of version 1.
 
You can think of a neighborhood as a collection of open balls, which is equivalent to many other definitions. For example, with regard to the two definitions you gave, a set containing an open set of x can be that open set of x itself. Remember, all sets contain themselves, by definition! In general, however, an open ball will be a proper subset of some neighborhood, which itself will be some open proper subset of the original set.
 

Similar threads

Undergrad On two definitions of locally compact
  • · Replies 5 ·
Replies
5
Views
3K
Graduate Understanding: double of conformally flat manifold is conformally flat
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Definition of inertia tensor from a differential geometry viewpoint
  • · Replies 15 ·
Replies
15
Views
2K
MHB Carothers' Definitions: Neighborhoods, Open Sets, and Open Balls
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Definition of Neighborhood, Very Confusing
  • · Replies 2 ·
Replies
2
Views
2K
Graduate About computing the tangent space at 1 of certain lie groups
  • · Replies 4 ·
Replies
4
Views
3K
Graduate Prove that points which are indistinguishable from 0 exist (using logic)
  • · Replies 17 ·
Replies
17
Views
2K
Undergrad Local Basis in Topology .... Definitions by Croom and Singh .... ....
  • · Replies 14 ·
Replies
14
Views
2K
Undergrad Is the Proof of Statement (3) in Lipschultz's Differential Geometry Justifiable?
  • · Replies 9 ·
Replies
9
Views
2K
Undergrad What is the difference between these two power series theorems?
  • · Replies 5 ·
Replies
5
Views
3K