What is Topology, and what's the big deal?

A finite set is strongly analogous to a bag of things. But all analogies are flawed and an infinite set allows for many possibilities that are never encounters in bags of things.

For example, if there is a mapping of everything in non-empty bag A onto the things in bag B such that that exactly two elements of A always map to the same element B, then it follows that the number of things in bag B is smaller than bag A. But this simple truth does not extend to infinite sets.

Consider \( f(x) = \textrm{max} \left( 1-x, \, x \right) = \frac{1 + \left| 2x - 1 \right| }{2} \) which maps the integers onto the positive integers and agrees with our bag-based intuition version \(g(x) = \left( -1 \right)^{\left\lfloor \frac{x+1}{2} \right\rfloor } \times \left\lfloor \frac{x+1}{4} \right\rfloor \) which maps the postive integers 2-to-1 with all the integers and therefore violently disagrees with a bag-based intuition. Making things worse, \(g(f(x)) = \left( -1 \right)^{\left\lfloor \frac{3 + \left| 2 x - 1 \right| }{4} \right\rfloor } \times \left\lfloor \frac{3 + \left| 2 x - 1 \right| }{8} \right\rfloor \) which maps four integers onto every integer.

A better notion of larger and smaller is needed for infinite sets than is provided by analogy with bags of stuff.

One of the great sea changes of mathematics happened when it was discovered that intuitive concepts of sets let to contradictions and so a formal theory of sets without those contradictions was created and much of mathematics was built on that foundation.

Cantor introduced a better notion of larger and smaller in 1870, which was just one step towards a robust set theory. The twentieth century saw vigorous development in the understanding of things (sets, groups, manifolds, etc) but also emerging was a growing importance of relationships between things and the resulting category theory is also advanced as a possible foundation for nearly all mathematics.

https://www.wolframscience.com/reference/notes/1127e
http://plato.stanford.edu/entries/settheory-early/
http://plato.stanford.edu/entries/category-theory/

 

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An aside.

Isn't there a large command in for TEX , I can't see some of rpenner's post.

 

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So topology is about " sets " inotherwords , statistics

 

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...Not as I under stand it. I.e. topology has ZERO concern for shape. E.G. a human and a doughnut have the same identical topology as both are "Doubly connected domains." I. e. from any two points on the surface, a & b, there are two topographic different way to connect them. I.e. the path can go on either side of the human's gut or the hole in the doughnut.

 

Billy T's Comment*...Not as I under stand it. I.e. topology has ZERO concern for shape. E.G. a human and a doughnut have the same identical topology as both are "Doubly connected domains." I. e. from any two points on the surface, a & b, there are two topographic different way to connect them. I.e. the path can go on either side of the human's gut or the whole in the doughnut.[/QUOTE]

River said :
Billy has become ....stupid

* Sorry when I try to reply there are wrong quotes made, and extra "[/quotes]" generated so I am force to be specific in the text. Yesterday, whole pages did not display. I think, but know little, that my ISP and the sciforums ISP have some "hand shaking" or indexing error?

 

Billy T said: " Not as I under stand it. I.e. topology has ZERO concern for shape. E.G. a human and a doughnut have the same identical topology as both are "Doubly connected domains." I. e. from any two points on the surface, a & b, there are two topographic different way to connect them. I.e. the path can go on either side of the human's gut or the hole in the doughnut."

No but this correction to arfa brane's OP is the first and only one I will ever be able to make as he knows much more than I on ANY math subject.

Perhaps the definition of topology has changed in the last 55 years to be more than just the study of how domains are connected?
Even if it has, I bet there still is no topographic difference between a human and a doughnut. Both have identical topology as both are "Doubly connected domains."

 

So is this about ego then ?

Billy t edited to make this a statement (question) by river. What displayed was as if I had said the above (at least on my screen.) I'm going to re-start my computer to see if that fixes the "what displays problem"

 

Topology is about "shape" in that, if you take for example a sphere (solid ball) of something plastic (plasticine!) and deform it, it will be topologically equivalent to a sphere (ball) as long as it isn't torn or cut. Hence any "blob" with a continuous surface and no holes is topologically equivalent to a sphere.
(What does change when you deform a sphere this way?)

Likewise, the "shape" of a knot doesn't change if you deform it by stretching parts of it, or looping the closed curve over itself. That is to say, the topological structure doesn't change. You could say that the topology isn't about "a" shape, but rather about changes in shape, so then it is about shape (isn't it?).

Then in Networks, like the one you're using, there is "network topology", which concerns both the physical layout (interconnections), and the "shape of signals" or logical topology.

 

No, it has been explained to you why this is not true - you require a metric for "shape" and, although the Hausdorff property describes separation, it is not a metric sufficient to describe "shape"

Wrong - the 2-sphere, say, is a 2-dimensional surface, the "solid ball" in this case is a 3-dimensional embedded structure whose boundary (if it exists) is the 2-sphere

See above why this is not true.

True, topology isn't about shape. The rest of your statement is opaque to me

Look, you have been told in some detail all you need to know for a working knowledge of topology (defined as the study of topological spaces). If you choose not to read what some of us went to some trouble to explain that is your privilege. But it is annoying that you pretend these efforts of ours where not made

 

So Prof Cochrane was misleading everyone? Why is a metric required, or why do you need distances to describe shape?
I'm sorry, but I don't really see why I should just accept what you've got to say, ok?
Here are several internet sources I googled.

I think some people have a somewhat fixed idea of what the word "shape" means. If topology is about changes in shape, then how is topology not "about" shape?
Why have I seen this so many times? By which I mean I've seen topology described as those properties of a space which do not change when the shape of the space changes.

What does a star topology "look like", does it have a shape?

p.s. I know a sphere is a surface, and I know what a (mathematical) ball is. That sentence I wrote is perhaps poorly phrased, I didn't make it clear that I meant two different things, so I can see how someone could interpret it as ignorance. But, I don't really care, you know?

 

Oh, boy. I am fixing to get some fire for this, because I have almost no math. I can see the things, though.

Topology is not about shape, it's about how shapes can be projected onto each other (sometimes), and what we can see when the projections are congruent or divergent that tells us about the surface (complex or not) and how it relates to other sets.

I have embarrassed myself probably, but I'm ignorant, so it doesn't matter very much.

To arfa, a star topology is a simple thing, most often used in ancient networking protocols, but yeah, it has a shape. It's flat and incompletely connected, depending on which node of the star is or isn't alive.

If this is basura, please ignore me. It's late and I've been drinking some very nice 12 year old Canadian.

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To Arfa Brane:
Surely you agree that a human and a doughnut have very different shapes, and that they both are doubly connected 3D domains with IDENTICAL TOPOLOGY, do you not?
Thus, it is best not to mention "shape" in the definition of topology, and in fact your first quoted definition specifically EXCLUDES any consideration of shape:

"noun: topology
1.1. Mathematics:
the study of geometrical properties and spatial relations unaffected by the continuous change of shape or size of figures."

The definition I learned ~55 years ago seem to still be valid.* I.e.
Topology are the same if they have the same max number of distinct paths between two points. For humans and doughnuts there are two distinct paths between points a & b on the surface or one or both in the material volume.

The second part of the definition would define "distinct." I.e. two paths are distinct if one can not be continuously deformed into the other while always remaining inside or on the surface of the object. (I just invented this definition, but think it is OK and complete at least for 2 or 3D objects.)

* Definitions even in science do change. Poor Pluto, is no longer a "planet" as it fails to clear other objects out of its orbit path.

 

Not all, he is saying exactly as I said all along

My apologies, I assumed you knew, for example, the definition of a circle as the set of points \(P(a,b) \in R^2\) with a fixed distance \(r\) from fixed point \(O \in R^2\) (this is of the course the embedded version)

By all means prefer these over those you have been given here.

But it raises the question - why ask a mathematical question on a science forum if you prefer the "googled" answers to those offered by qualified members on that forum? To say the least, it is ungracious to suggest your respondents are thereby wrong

 

This ^^ is what QuarkHead has been saying all along.

A circle has a fixed radius, as Quarkhead has noted.
What about a square, or any closed non-intersecting curve? These are topologically equivalent to a circle.
How does topology "know" a square and a circle (regardless of the length of its radius), are equivalent? How do you change a square into a circle, or into any closed curve, without changing the shape of the square?

Let's go with the idea that topology is not about shape; then how can it be about changes in . . . shape? That's the "naive" question for today, folks.

 

Hmm. By using the term "circle" you are assuming the metric I hinted at. So I am not sure what you mean. Likewise the term "square".

However, to go with the flow - suppose a metric. Then the notions of "circle" and "square" as shapes make sense (let us say)

Now let us see you define a bijection from a vertex of the square to a point on the circle. I don't believe it can be done.

I can't be bothered doing the mathematics this late, but try to falsify my assertion

That, by your own (implicit) admission is a non sequitur

 

I don't know why you're being so obscure, or why you think I'm assuming anything. A circle has a fixed radius, a circle of a given size has a fixed length radius.

But circles and squares are topologically equivalent, as I'm sure you know, because both separate the plane into two regions. This is true for any closed curve, which I'm sure you also know. And the topology is not dependent on a notion of distance (a metric), perhaps you also know this? In graphs, the topology is defined by how vertices are connected to other vertices, not the distance between vertices or the length of edges (but you knew that, right?).

As for "shape", it's obvious to a child that a circle and a square aren't the same shape. What's the problem here?

 

So I've been thinking of an easy way to show that the square and the circle are homeomorphic.

"The" square and "the" circle means any square and any circle; topology isn't about the size of mathematical objects, it's about whether two objects which have a different geometry (for which the circle and square are obvious candidates), are equivalent in a more general way, and need not have a notion of distance defined.

So place one inside the other (i.e. choose one smaller than the other and align their centres, since any choice is valid). Then any ray projected from the centre outwards will intersect each "curve" at exactly one point on each.

Knots don't have kinks in them like squares do, and squares aren't continuously differentiable (they aren't smooth like a circle), if a knot is kinked, you just unkink it, type of thing.

 

For 1-to-1 mapping between the square \((0,1)\times(0,1)\) and the interior of the unit disc, one possible mapping is

\( \begin{pmatrix} x' \\ y' \end{pmatrix} = \frac{ \textrm{max} \left( | x - \frac{1}{2} | , \, | y - \frac{1}{2} | \right) }{ \sqrt{\left( x - \frac{1}{2} \right)^2 + \left( y - \frac{1}{2} \right)^2}} \begin{pmatrix} 2 x - 1\\ 2 y - 1 \end{pmatrix} = 2 \frac{ \left| \begin{pmatrix} x - \frac{1}{2} \\ y - \frac{1}{2} \end{pmatrix} \right|_{\infty} }{ \left| \begin{pmatrix} x - \frac{1}{2} \\ y - \frac{1}{2} \end{pmatrix} \right|_{2} } \begin{pmatrix} x - \frac{1}{2} \\ y - \frac{1}{2} \end{pmatrix} \)

 

The business of centering one of the square or circle inside the other isn't necessary, since the smaller object can in fact (in general) be anywhere in the interior of the larger.

Then any interior point of the smaller can have this projection (a ray) directed outwards intersecting both curves, in any direction (a ray will intersect exactly one point on the boundary of each closed curve). So it's a matter of choosing a square (resp. a circle) whose boundary is a proper subset of the interior of a circle, then choosing a point in the interior of the square (resp. circle) for the origin of any projected ray.

 

Ok, the projection argument only works with convex figures, so it doesn't generalise. If a closed, not necessarily convex kind of curve is drawn, then you want to deform it so it is convex, say. You want to give it the least number of bends.

An arbitrarily drawn, closed non self-intersecting plane curve will be described by a fairly complex function of x and y, given some choice of coordinate axes in \( R^2 \). But you can see that there is a continuous function that deforms such a curve into a circular one (you can "prove" this physically with a loop of string).

A circle is a special kind of closed curve because it's the locus of points equidistant from a central point--the centre has a particular status for a circle; a square or a triangle (or any n-gon) doesn't have this property.

As for a bijection between the vertices of a square and points on a circle, that's what the projection "method" is (it isn't an original idea, I just thought it up too), it shows that since say, a small square can be anywhere in the interior of a large circle, each vertex of the square is projected to an arbitrary point on the circle, and we are done.