Undergrad How do I use the four axioms of a neighborhood to define an open set?

  • Thread starter Thread starter learning physics
  • Start date Start date
  • Tags Tags
    General topology
Click For Summary
An open set can be defined using the four axioms of topological neighborhoods, which assert that each point in the set has a neighborhood contained within the set. The intuitive understanding of an open set is that it allows for movement around each point without reaching a boundary. The discussion emphasizes the concept of "The Big Neighborhood," where each point is surrounded by infinitely smaller neighborhoods. However, there is confusion about how this formal definition aligns with the intuitive notion of openness. Clarification is sought on the relationship between the axiomatic definition and the intuitive concept, along with a request for a reference link.
learning physics
Messages
2
Reaction score
0
TL;DR
How do I use the four axioms of a neighborhood to define an open set?
How do I define an open set using only the four axioms of topological neighborhoods, as per the Wikipedia article on topological spaces?

The intuitive definition of an open set is that it's a set of points on a real number line containing only points at which there is room for some hypothetical point-sized particle to move on either side.

I can see that an open set is defined as a neighborhood of all of its points, but how does this fit with the intuitive definition?

Suppose that we call a set of points that acts as a neighborhood of all of its points "The Big Neighborhood." Each point in The Big Neighborhood is contained in a neighborhood that is contained in The Big Neighborhood, which we'll call "smaller neighbrohoods." Each point in The Big Neighborhood is contained in a neighborhood that is contained in a smaller neighborhood. And so on.

So, I can see that each point in a neighborhood of all of its points is buried inside an infinite nest of smaller and smaller sets. But I don't see how this fits with the intuitive definition. Can anyone help?
 
Physics news on Phys.org
You better include a link to the reference that defines the four axioms you are talking about. I can't find what you are talking about in Wikipedia.
 

Thread 'About the definition of topological manifold using closed sets'

We all know the definition of n-dimensional topological manifold uses open sets and homeomorphisms onto the image as open set in ##\mathbb R^n##. It should be possible to reformulate the definition of n-dimensional topological manifold using closed sets on the manifold's topology and on ##\mathbb R^n## ? I'm positive for this. Perhaps the definition of smooth manifold would be problematic, though.
View full post »

Similar threads

Undergrad How to define an open set using the four axioms of a neighborhood
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad On two definitions of locally compact
  • · Replies 5 ·
Replies
5
Views
3K
High School Topological neigborhoods that are not intervals in the real line?
  • · Replies 15 ·
Replies
15
Views
2K
Undergrad Why is [0,1] compact in the case of a collection of open neighborhoods?
  • · Replies 9 ·
Replies
9
Views
5K
MHB Carothers' Definitions: Neighborhoods, Open Sets, and Open Balls
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Defining Neighborhoods in Topology: Inclusion vs. Containment
  • · Replies 12 ·
Replies
12
Views
3K
Graduate Regarding fibrations between smooth manifolds
  • · Replies 1 ·
Replies
1
Views
2K
Understanding the Open Set: Definition & Examples
  • · Replies 19 ·
Replies
19
Views
4K
Undergrad Local Basis in Topology .... Definitions by Croom and Singh .... ....
  • · Replies 14 ·
Replies
14
Views
2K
Undergrad Can a Topological Space be Both a 1-Manifold and an n-Manifold?
  • · Replies 17 ·
Replies
17
Views
3K