What is topology, and what's cool about it?

Factual Questions
61

Yeah, you’re right. Though I have to say that my current academic (administrative) work involves much too little topology for my taste.

You didn’t think I could stay away from this thread for very long, did you?

Before I give my thoughts about the OP, let me throw my $0.02 into the circle argument. I haven’t thought through the whole thing, though my intuition is that you won’t be able to find anything quite like a circle in general. However: If you’re willing to deform your temperature function just a little, you certainly can get a circle. Consider the function that takes every point on the sphere to the difference between the temperature at that point and the temperature at its antipodal point. This is continuous and, assuming temperature isn’t constant, takes on both positive and negative values. By modifying the temperature function just a little, we can make this function differentiable and transverse to 0. Whatever that means, it has the consequence that the set of points on the sphere that get mapped to 0 forms a one-dimensional manifold, which must be a union of circles. I believe you can then show that exactly one of these circles must have the property that, for each point it contains, it must also contain the antipodal point. That’s the circle you want.

Even if all that’s true (and there’s too much handwaving at the end), I don’t see that it says anything nice about a general continuous temperature function. Starting from a nice temperature distribution that does give you a circle and deforming it can do some pretty nasty things to that circle.

62

I certainly accept the final result, being aware that it can be proven by different means. As for the sub-result (essentially, “For any odd function from the surface of a sphere to the real line, there is some path from a point to its antipode along which this function is constantly zero”), I am not personally aware of any particular reason to consider it true. It may well be true, it may well be false, so far as I know. So, I suppose, in some sense, I do dispute that, though not in the strong sense of actually believing it to be false, but merely the weak sense of not being convinced it’s true.

63

Some thoughts on what topology is: It’s several things.

As Indistinguishable said, at its heart, and as a foundational element of many other parts of mathematics, topology is about continuity. Continuity is a property of a function from one set to another. To abstract the idea of continuity it turns out that the sets should have a certain kind of structure, called “a topology,” defined on them, and the functions should respect that structure in a certain way. I’m avoiding the details of the structure not because it’s complicated – it’s quite simple and elegant – but because it takes some convincing to connect it to intuitive ideas of continuity.

But then it also turns out that putting a topology on a set determines its shape in a certain loose sense: the sense in which a donut and a coffee cup have the same shape. A set with its shape is called a topological space. Topology is also about the interplay between spaces and continuity. Properties of spaces determine properties of functions defined on them; the intermediate value theorem has to do with the connectedness of a space, while the extreme value theorem has to do with the compactness of a space (these things having precise definitions). The Borsuk-Ulam theorem being mentioned is another example. Conversely, the possible continuous functions into or out of a space tell you a good deal (or everything, from the right point of view) about the space. For example, you can draw a circle on the surface of a donut (i.e., a continuous function from a circle to the donut) that can’t be continuously deformed to a point, which is something you can’t do on the surface of a sphere. So, in some parts of topology, including my own, the spaces become the primary objects of study, and continuous functions a tool in that study.

It’s getting late and I’m rambling more than I had hoped. I’ll leave it at this for the night.

64

Nope, but sounds like you’re a GIS person too. I can certainly understand topology in that context, but when you get to the higher math of it, hoo boy.

I’ve never heard the term “face” used in GIS context. I have an idea, but can you expound?

65

Aak, never mind. TINs.

66

My general thinking is to take a bunch of longitude great circles (that is, great circles which pass through the poles). For each of those great circles, you can (by the IVT) find a pair of antipodal points with the same temperature. Now, take the set of all those points, for each of those great circles. That set of points must be one-dimensional, and I think continuity of the temperature function guarantees that it can’t have endpoints, so it must be a circle. And it must have at least one point on each longitude line, so it girds the Earth.

Yes, this still isn’t rigorous (in particular, I’m handwaving a bit on the no-endpoints bit), but I think that’s an outline of how I’d go about proving it.

67

Say I have a pair of antipodal points A and A’ that are the same temperature but A is a local maximum and A’ is a local minimum (w.r.t. the temperature function.) In this case A (nor A’) is not path connected to another point that that is the temperature as its antipode.

Now consider two arcs of a a great circle L and L’ chosen such that every point on L is the antipode of a point on L’. Now assume the temperature along both L and L’ are constant and identical to each other. Now assume we have open sets containing each of L and L’ such that the closure of the set containing L has a minimum that is achieved only by points on L and the closure of the set containing L’ has a maximum only achieved by points on L’. This gives us a line (of same temp antipodal points) with endpoints even with a continuous temperature function.

68

Re: Chronos

You acknowledge that isn’t rigorous, but, just to highlight the particular things which would trouble me most:

When you say “that set of points must be one-dimensional”, what exactly do you mean by this? Certainly, one could imagine that every point was equi-antipodal, so that the entirety of equi-antipodal points formed a two-dimensional surface. If you mean “Choose (in some nice way) one equi-antipodal point from each line of longitude, and then consider the set of such chosen equi-antipodal points”, the question remains as to what constitutes a “nice” choice (e.g., a suitably continuous one, in some sense) and how we know that we can always make one.

What does “girds the Earth” mean? If it just means having a point on each line of longitude, then we will have that every line of latitude, and, indeed, each pole by itself girds the Earth, but in this case, merely girding the Earth will not, in itself, suffice to buy us everything we need. To use the usual IVT trick, we need to know that we can traverse a path from one point to its antipode, which certainly won’t happen with general lines of latitude.

It’s also certainly possible to have some equi-antipodal points which are “endpoints”; e.g., one could imagine a distribution that did whatever continuous thing one liked north of the tropic of cancer, smoothly tapered off to 0 around the Equator, and then took on at every southern point the negation of its value at the corresponding northern antipode. The equi-antipodal points on this distribution would be the zeros, which, while including the Equator, could also contain all kinds of arbitrary figures north of the tropic of cancer, with all kinds of endpoints within these. So, again, there is some question about making the “right”, “nice” choices of equi-antipodal points from each line of longitude, not just arbitrary ones, and how we know we can do so.

Anyway, you acknowledged that this was just a rough, handwaving argument-outline so far, and it might still be fleshable out to a proof in some development. But these are the main concerns which raise their heads to start with.

69

Eh, I fear my intent with the above may be misinterpreted as well. My intent isn’t to slam Chronos’s attempt or any such thing; my intent is merely to highlight where I am worried that it will not ultimately work out, and thus where I’d like to be able to see holes filled in. I’d love to have a proof of this result from little more than the IVT, but I’m worried that one can’t be given.

70

:v: The Seventh Deadly Finn likes this.

71

After further reflection I have determined that a great circle can exist with exactly one pair of equal temp antipodal points where one is a local min and the other a local max with a continuous temperature function.

I’ll try to describe it but if this is unclear I can work up a diagram or something…

We can represent the temp function using a (star convex) closed loop that contains the origin, where the angle represents our position on a great circle and the distance from the origin measure temperature. (Use kelvin or something so everything is nice and positive).

The loop in question will be shaped sort of like a heart (Valentine not anatomical) with point of the heart and the point between the humps the same distance (call this r) from the origin. These are our equal temp antipodal points.

Draw a circle of radius r around the origin and this intersects the heart at four points. Assume these points are at 60, 90, 120, and 270 degrees. The 90 and 270 degree points are our previously mentioned pair.

Draw lines from the other two points to the tip of the heart. Push the curve that defines the bottom left of the heart almost all the way out to the circle of radius r and deform the of side so it is a straight line.

Now we have what we were after.

72

Like this…

http://www.unc.edu/~schneidk/heart.jpg

73

Sorry about the triple post but now I’ve realized the previous two posts are in error.

The picture described and linked above does indeed contain at least one other pair of antipodal equal temp points.

74

Well, Lance, whether you’re right or not, that’s a good example, I think, of what my original question was about. If you’ve got a problem that’s kind of shape-y, you can use topology, the science of (among other things) making true mathematical statement about shape-y subjects, to solve it mathematically, rather than having to make a bunch of models out of cardboard.

As remedial as that sounds, it seems pretty cool to me that one can write equations and functions about shapes that turn out to be true. To a non-math person, that’s sort of amazing. I’ll bet it’s faster and more accurate than these messy, repeatedly-erased pencil diagrams I’ve been making to try to follow this conversation, too. :slight_smile:

75

Maybe later tonight or tommorow I’ll outline my “proof”. Keep in mind its a physics/graph type proof, not a rigorous math type proof.

If you read my posts carefully for intent and concepts, the way to do its all there.

76

There are a lot of non-rigorous proofs that look great, but break down when you actually start to examine the details. That’s the whole reason why we have standards of rigor. I hope you’ve got something, but if you’re going to convince the mathematicians, you’ll need something more than intent and concepts.

77

And your contribution to this thread is what ?

Least I got some damn ideas.

As I understand it, Chronos’s claim about antipodal points that have equal temp and pressure is an accepted “fact”.

Its not like I am claiming I’ve discovered the proof to the last digit of pi and am going nobel with it.

Chronos put something out there, some folks said "hows that ? ", even Chronos aint claiming a publishable proof here.

I’ve got an explaination that works for me. Not to say I wont find a fatal flaw along the way.

And bump the mathematicians. Ima physics dude, not nature’s bookeeper.

78

billfish678, I believe you are again taking remarks as far more personal than they are intended to be…

79

Topology is great, but it is a veritable hydra of areas now.

If you take a beginning course in topology, you basically get a calculus of sets. topics like openness, metrics, continuity, limits, filters …

If you keep going, you get more heads: Point Set Topology, Algebraic Topology, Knot Theory, …

80

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