What is the relationship between points and neighborhoods in topology?

[StatusX]

You seem to be saying that because the distance between any two points is finite, there can't be a continuum because there would have to be a distance between consecutive points. The flaw in this argument is that there arent any consecutive points in a continuum. This is counter-intuitive, but not illogical. Between any two points you pick there are still an infinite number of points, regardless of how close they are. There is no next number after 1. The open set (0,1) has no greatest or least element.

 

[Hurkyl]

A brief introduction to topology.

A topology consists of two kinds of things:
(1) points
(2) neighborhoods

The basic relationship between points and neighborhoods is that neighborhoods contain points. In fact, in the set theoretic approach to topology, neighborhoods are defined to be the set of all points they contain.

Furthermore:
Each point is contained in at least one neighborhood.
If two neighborhoods overlap (that is, have a point in common), then there is an entire neighborhood contained in both.


One example of a topology is the real line. The points of the real line are simply real numbers. The neighborhoods of the real line are the open intervals: that is, sets of the form {x | a < x < b}. for some a and b.


Non-mathematical aside: The points, by themselves, tell you very little. The neighborhoods are the "soul" of topology -- they are what describes how the points relate to each other, they describe "texture" of the topological space. As we see with the example above, the neighborhoods of the real line are precisely the neighborhoods Canute mentioned. I don't think that's a coincidence: Canute wasn't the first person to realize that these ranges are important to describing a "continuum".


Back to the mathematics.

Another type of example of a topology is a discrete space:

The points can be anything (but, IIRC, there's supposed to be at least 1).
Then, for each point, there is a neighborhood that consists of that point and nothing else.

Each point in a discrete space is isolated: for each point there is a neighborhood that contains that point and nothing else.

Contrast this with the real line: every neighborhood of a point contains many other points.


Next, I'd like to mention the notion of nearness. If you have a point (let's call it P), and you have some set of other points, (let's call it A), then the phrase P is near A means that every neighborhood of P contains a point in A.


Let's use the real line again as an example. Let's let P be the point 0, and let A be the set {1, 1/2, 1/3, 1/4, ...}. Then, P is near A.

Proof: Let (a, b) be any neighborhood of P. That means a < 0 < b. However, there exists some integer n such that 1/n < b, which means that 1/n is in the neighborhood (a, b). QED


Note that the intuitive notion of a "gap" can now be described in terms of nearness -- no need to have any concept of there being some other locations that make up the gap. We can say there's a gap between a point P and a set of points A if P is not near A.

So, we can see that in the discrete space, there is a gap between a point and any set not containing that point! However in the real line, there is no gap betwen 0 and {1, 1/2, 1/3, 1/4, ...}. But, of course, there is a gap between -1 and {1, 1/2, 1/3, 1/4, ...} (because the neighborhood (-1.5, -.5) doesn't contain any element of the set)

And, just as we'd expect, there is a gap between any two points on the real line: for instance, there's a gap between 0 and {1} because the neighborhood (-0.5, 0.5) doesn't contain any element of {1}.


There's obviously a lot more to say. I haven't even gotten far enough that we could start speaking about what it means to be a "continuum". But, I was just trying to give a taste about how one can speak of a space being made up of individual points without them being necessarily isolated.