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

[learning physics]

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?

 

[FactChecker]

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.

 

[PeroK]

This thread is a duplicate of:

https://www.physicsforums.com/threa...ing-the-four-axioms-of-a-neighborhood.1059241