What is the relationship between points and neighborhoods in topology?

[Canute]

The point is the location. The location is the point.

Does it make any sense to talk about two locations at the same location?

No, two locations are clearly different locations if they have been defined as such. I don't mind whether we call them points or locations. We have defined them as being the same thing.

That would be precisely the same thing as trying to claim that there are two or more points at the same location. If they are at the same location then they are the same point.

Ok. And if they are the same point then they are at the same location.

Don't think of points as being little entities like sub-atomic particles. They are locations. Period amen. They aren't physical entities.

I understand that. These are points located in our imagination.

Once again you are getting way ahead of the game here. Coordinate systems are nothing more than systematic ways of labeling locations. Once you have the concept of a coordinate system you can have all the locations you like.

But only if your coordinate system is infinitely finely grained.

The ultimate restriction is that if your coordinate system is a field of locations, and locations are nothing more than dimensionless points in that system, then you are stuck with our original philosophical conclusion that there must necessarily be gaps between these locations. In other words, you're coordinate system (if it is to display a quantitative nature) must necessarily be a quantum field.

I didn't mean to say anything much about coordinate systems. I was just pointing out that two locations imply a coordinate system.

I submit to you that if the universe really was a continuum it would not display the quantitative nature that our universe displays. It simply isn't possible to talk about more than one location in a universe that is a continuum.

That was Parmeneides' and Zeno's point, and many others. The question is perhaps, what meaning can points and locations have outside of the coordinate system we call spacetime. As far as we can tell spacetime, our universe anyway, has not always existed, but exploded into being just as if the BB happened at every point in it at once.

Actually that's a very profound bit of philosophy right there and you may very well be correct. Our entire universe may not take up any 'space' at all actually. In fact, on the deepest philosophical level I wouldn't be a bit surprised if that isn't the 'true' nature of our existence.

Yes, this is the fundamental issue. Really we're talking about the nature of the one and the many, and back with Plato et al.

However, if you want to think that deeply then consider this. A universe that is a continuum doesn't really have any space at all between any of its points so it wouldn't require any space to exist in its entirety either! :scream:

Exactly. What could it mean to say that the universe takes up space? The idea makes no sense.

But getting back to the logic. You keep wanting to put more points into the gap between two points. [/quote}
I don't want to put them in. It just follows from the fact that points are defined only by their location that there must be points between different points. It's just a consequence of the definition.

But if you go back to the discussion of the preceding quote above our logic showed us that there must necessarily be a smallest gap between two points which cannot be thought of as being divided up further. In other words, by pure logic, there necessarily must exist some smallest gap where it is simply meaningless to talk about inserting more points. That was the whole purpose of the premise of thinking about what it would mean to talk about only TWO dimensionless points that are not the same point.

I'm sorry but I cannot conceive of a gap so small that an infinitessimal wouldn't fit into it. It's possible to define gaps in such a way as to stop me from doing this, for practical or formal reasons, but you can't reify a definition.

Oh,… and the existence of points does imply a coordinate system. Imagine a coordinate system that contains only two points. You're either on point A or point B. That's the entirety of that coordinate system. There's simply no other place to be. If point A and point B are the only two points that exist. And they are not the same point. Then you are either on point A or point B, but it's meaningless to talk about being half-way between them.

That seems self-contradictory, but I may be misreading it.

And more to the point (not pun intended), if the points are dimensionless and they are not the same point then they must be two different locations (because that's what a point is) and they cannot be touching because they are dimensionless. So there is no other conclusion to come to except that they are separate locations with no other location permitted to exist between them. Thus the notion of a gap where no points can logically be said to exist.

Well, there is at least one other conclusion, and that is that your definition of points, locations and gaps is incoherent. Btw, I'm not trying to defend some particular theory here, I simply can't see how you arrive at your conclusions.

 

[NeutronStar]

It's not a personal limitation. It's a logical contradiction.

I'm with Hurykl on this one. There seems no reason that your inability to conceive of a continuum should have any bearing on whether or not the calculus can be used to model one. The question seemed sensible to me, and I thought the answer would be well known. If you don't know the answer then on what basis are you arguing that the number line cannot be considered as a continuum? Because you can't imagine it?

Ok, this was due to poor phrasing on my part. When I say that I can't imagine it I mean that I can't logically justify it. I don't mean to imply that I have a limited imagination.

In other words, here is what we have to imagine in order to "justify" the idea of a continuum.

We begin with the fact that points are dimensionless. Remember, if we claim to have points that have dimension then we have merely shifted the discrete gap into the points and we haven't really solved the problem.

So, the points must be dimensionless.

Ok. So far so good. I can conceive the idea of a dimensionless point as simply the idea of a location that has not breadth.

Now, we need to consider what it means to have two distinctly different points that are not the same point. (in other words, two distinct locations that are not the same location)

But wait, we're not done! We have to maintain the concept of a continuum which is the idea there there are no gaps between these distinctly different dimensionless points.

Well, how are you going to envision that? In other words, how are you going to logically justify that concept?

When I say that I can't envision it, I simply mean that it is a logical contradiction. I maintain that it cannot be logically justified. And in that sense I cannot conceive it as a meaningful idea. It's illogical.

The idea of two distinctly different locations that are located at the very same location (which they must be if these locations have no breadth, i.e. are dimensionless points) is simply a logical contradiction plain and simple.

So I probably shouldn't just feebly claim that I can't envision it. I should boldly claim that it is a logical contradiction and therefore it is nonsensical.

How can anyone claim to have an idea that cannot be conceived?

The idea of a continuum is a logical contradiction pure and simple.

This has absolutely nothing at all to do with my own personal inability to imagine it. I can clearly see what it would take to try to imagine it and I see that it is a direct contradiction in logic. We simply can't claim that two distinct dimensionless points are not the same point. It is a logical contradiction. Its an unworkable idea!

This has absolutely nothing at all to do with my own personal abilities to comprehend anything. I claim that anyone who believe that they can comprehend this idea if necessarily fooling themselves.

I would be more than happy to hear arguments to the contrary. But so far no one has offered any logically consistent picture of how a continuum can logically be said to exist even as an idea.

Yet, I have offered a logically consistent picture of what it means to have two discrete points. So I see this position as being more meaningful.

 

[matt grime]

What do you think I am? A crackpot?

Do you really want an answer to that?

Right now it seems to me that many mathematicians are confused about the actual property of cardinality between the set of real numbers and the set of natural numbers so,…

We are confused? Pray tell what the correct definition is? Simply that they are infinite? Well, that's a very old fashioned view that we can *refine*.

Can you tell me in clear intuitive terms what this difference is?

Why must it necessarily be intuitive? In the category of SET they lie in different isomorphism classes. That's all.

Does the set of real numbers actually contain "more" elements than the set of natural numbers?

that's up to you to state what you mean by "more" isn't it? By *analogy* with the case of finite sets, we could say Y has strictly more elements than X if there is an injection from X to Y, but no bijection, that is X is in 1-1 correspondence with a proper subset, but never the whole of Y. That seems a reasonable generalization of "more" doesn't it, I suppose.

With it, we can say seemingly natural statements such as there are real numbers that are not algebraic, since there are strictly more real numbers than algebraic ones. However, that is obscuring the simple fact that algebraic numbers are countable and Reals not.

Or is the cardinal difference between these two sets based on a different quality other than quantity?

no it is to do with the isomorphism class in SET, nothing more nor less.

If you answered the previous question "yes:, then just what is this other quality? What makes these sets cardinally different from each other if one does not contain more elements than the other?

i didn't answer "yes", or perhaps you think I did. Whatever, the point is that the only person who appears not to know what cardinals are is you.

 

[Canute]

This bothers me because I'm interested in science and discovering the true nature of our universe. Yet modern science is becoming almost entirely mathematical and less experimental. So if mathematics doesn't reflect the true nature of quantity, and science is becoming more mathematical, then eventually science will fail to describe the universe.

I agree with your diagnosis, but not with the cure.

In any case, I'd like to hear your thoughts on anything that I said here, and also any ideas that you might have on trying to describe the nature of two distinct points in a continuum. I can't even begin to imagine trying to intuitively describe the notion of two dimensionless points in a continuum. How would one go about conceiving of such an idea?

I also agree that this is an inconsisistent idea. The only points there can be in a continuum are conceptual ones.

 

[Canute]

We begin with the fact that points are dimensionless...(snip) ...So, the points must be dimensionless.

Hmm.

Ok. So far so good. I can conceive the idea of a dimensionless point as simply the idea of a location that has not breadth.

It's the only sort of dimensionless point there is.

Now, we need to consider what it means to have two distinctly different points that are not the same point. (in other words, two distinct locations that are not the same location)

But wait, we're not done! We have to maintain the concept of a continuum which is the idea there there are no gaps between these distinctly different dimensionless points.

Why? Your points are in your imagination, you won't find any out there in reality.

The idea of two distinctly different locations that are located at the very same location (which they must be if these locations have no breadth, i.e. are dimensionless points) is simply a logical contradiction plain and simple.

I agree.

The idea of a continuum is a logical contradiction pure and simple.

I more or less agree with that also. However I don't derive the same conclusions from it.

This has absolutely nothing at all to do with my own personal inability to imagine it. I can clearly see what it would take to try to imagine it and I see that it is a direct contradiction in logic. We simply can't claim that two distinct dimensionless points are not the same point. It is a logical contradiction. Its an unworkable idea!

I don't think anyone has claimed that.

But so far no one has offered any logically consistent picture of how a continuum can logically be said to exist even as an idea.

That's a fair point, but I won't respond here.

 

[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.