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

In summary, an open set can be defined using the four axioms of a neighborhood, which include: (1) Each point in the set must have a neighborhood; (2) The neighborhood must be contained entirely within the set; (3) The union of neighborhoods must also be an open set; and (4) Finite intersections of neighborhoods must also form neighborhoods. Together, these axioms establish the properties that characterize open sets in a topological space.
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learning physics
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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?
 
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You could try this guy's YouTube series:

 
  • #3
learning physics said:
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: [itex]U \subset \mathbb{R}[/itex] is open if and only if for each [itex]x \in U[/itex] there exists [itex]\delta > 0[/itex] such that [itex](x - \delta , x + \delta) \subset U[/itex]. (You can move up to [itex]\delta[/itex] away from [itex]x[/itex] in either direction without leaving [itex]U[/itex].)

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.

[itex](x - \delta, x + \delta)[/itex] is a neighbourhood of [itex]x[/itex].
 

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

What are the four axioms of a neighborhood used to define an open set?

The four axioms of a neighborhood used to define an open set are: (1) For every point in the space, there exists a neighborhood around that point. (2) The neighborhood of a point contains points that are "close" to it. (3) A neighborhood can be defined in terms of a radius, meaning that for each point, there exists a positive radius such that all points within that radius belong to the neighborhood. (4) The intersection of two neighborhoods is also a neighborhood, ensuring that neighborhoods can be combined.

How does the concept of a neighborhood relate to open sets?

A neighborhood around a point is a set that contains points "close" to that point. An open set can be defined as a collection of points where, for every point in the set, there exists a neighborhood fully contained within the set. This means that you can move a small distance from any point in the open set and still remain within the set itself.

Can you provide an example of an open set using the neighborhood axioms?

Consider the open interval (a, b) in the real numbers. For any point x in this interval, you can find a neighborhood by choosing a radius r small enough such that the interval (x - r, x + r) is entirely contained within (a, b). This demonstrates that every point in (a, b) has a neighborhood that lies entirely within the open set, satisfying the definition of an open set.

Are all neighborhoods open sets?

Not all neighborhoods are open sets, but every open set can be thought of as being made up of neighborhoods. A neighborhood can be defined in a broader sense and may include boundary points, while an open set specifically requires that all points within it are interior points, meaning that neighborhoods around those points do not include any boundary points.

How do the axioms of neighborhoods help in understanding topology?

The axioms of neighborhoods provide a foundational framework for understanding the concept of open sets in topology. They help to establish the basic properties of continuity, convergence, and compactness by allowing mathematicians to rigorously define concepts of closeness and limits. This understanding is crucial for exploring more complex topological spaces and their properties.

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