How to define an open set using the four axioms of a neighborhood

[learning physics]

TL;DR Summary

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

I am struggling to define an open set using the four axioms of a topological neighborhood, as per the Wikipedia article "Topological spaces."

An open set on a real number line is a set of points that contains only interior points, meaning that there is always room for some hypothetical particle to move either side of each point. Let's call this "the intuitive definition" of an open set.

An open set is defined as a neighborhood of all of its points, but I don't see how that would connect to the intuitive definition.

If a set of points is a neighborhood of all of its points, which we'll call The Large Neighborhood, it means that each point is contained in a neighborhood even smaller than The Large Neighborhood, which we'll call "small neighborhoods." Each point in The Large Neighborhood must also be contained in a neighborhood even smaller than the small neighborhoods, and so on. So, each point in The Large Neighborhood is buried underneath an infinite number of neighborhoods that are smaller than The Large Neighborhood. Still don't see how this connects to the intuitive definition.

Anyone care to help?

 

[PeroK]

You could try this guy's YouTube series:

 

[pasmith]

TL;DR Summary: How do I use the four axioms of a neighborhood to define an open set?

I am struggling to define an open set using the four axioms of a topological neighborhood, as per the Wikipedia article "Topological spaces."

An open set on a real number line is a set of points that contains only interior points, meaning that there is always room for some hypothetical particle to move either side of each point. Let's call this "the intuitive definition" of an open set.

State this formally: $U \subset \mathbb{R}$ is open if and only if for each $x \in U$ there exists $\delta > 0$ such that $(x - \delta , x + \delta) \subset U$. (You can move up to $\delta$ away from $x$ in either direction without leaving $U$.)

An open set is defined as a neighborhood of all of its points, but I don't see how that would connect to the intuitive definition.

$(x - \delta, x + \delta)$ is a neighbourhood of $x$.