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- 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?
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?