So far, we’ve been talking about topologies in the most general sense: point-set topology. As we’ve seen, there are a lot of really fascinating things that you can do using just the bare structure of topologies as families of open sets.
But most of the things that are commonly associated with topology aren’t just abstract point-sets: they’re *shapes* and *surfaces* – in topological terms, they’re things called *manifolds*.
Informally, a manifold is a set of points forming a surface that *appears to be* euclidean if you look at small sections. Manifolds include euclidean surfaces – like the standard topology on a plane; but they also include many non-euclidean surfaces, like the surface of a sphere or a torus.
The formal definition is very much like the informal, with a few additions. But before we get there, I’ll refresh your memories about some concepts that we’ll need.
- A set **S** is *countable* if/f there is a total, onto, one-to-one function *f : **S** → **Z**;*, (where **Z** is the set of natural numbers), mapping each element of **S** onto exactly one natural number.
- Given a topological space (**T**, τ), a *basis* β for τ is a collection of open sets from which any open set in τ can be generated by a finite sequence of unions and intersections of sets in β.
- A topological space (**T**, τ) is called a *Hausdorff* space if/f if for any two distinct points in T, each point is a member of *at least* one neighborhood that the other is not.
A topological space (**T**,τ) is an *n*-manifold if/f:
- τ has a *countable* basis.
- (**T**,τ) is a Hausdorff space.
- Every point in **T** has a neighborhood homeomorphic to an open euclidean *n*-ball.
Basically, what this really means is pretty much what I said in the informal definition. In a euclidean *n*-space, every point has a neighborhood which is shaped like an *n*-ball, and can be *separated* from any other point using an *n*-ball shaped neighborhood of the appropriate size. In a manifold, the neighborhoods around a point *look like* the euclidean neighborhoods.
If you think of a large enough torus, you can easily imagine that the smaller open 2-balls (disks) around a particular point will look very much like flat disks. In fact, as the torus gets larger, they’ll become virtually indistinguishable from flat euclidean disks. But as you move away from the individual point, and look at the properties of the entire surface, you see that the euclidean properties fail.
Another interesting way of thinking about manifolds is in terms of *charts*, and charts will end up being important later. A *chart* for an manifold is an invertable map from some euclidean manifold to *part of* the manifold which preserves the topological structure. If a manifold isn’t euclidean, then there isn’t a single chart for the entire manifold. But we can find a *set* of *overlapping* charts so that every point in the manifold is part of *at least* one chart, and the edges of all of the charts overlap. A set of overlapping charts like that is called an *atlas* for the manifold, and we will sometimes say that the atlas *defines* the manifold. For any given manifold, there are many different atlases that can define it. The union of all possible atlases for a manifold, which is the set of *all* charts that can be mapped onto parts of the manifold is called the *maximal* atlas for the manifold. The maximal atlas for a manifold is, obviously, unique.
For some manifolds, we can define an atlas consisting of charts *with coordinate systems*. If we can do that, then we have something wonderful: a topology on which we can do angles, distances, and most importantly, *calculus*.
Topologists draw a lot of distinctions between different kinds of manifolds; a few interesting examples are:
- A *Lie group* is a manifold with a valid closed *product* operator between points in the manifold.
- A *Reimann* manifold is a manifold on which you can meaningfully defined angles and distance.
- A *differentiable* manifold is one on which you can do calculus.
Some errors:
1) Open sets are arbitrary unions of base elements, not just finite.
2) What you defined as a Lie group is actually an H-manifold, and much less than a Lie group. The “next” object up would be a topological group which is a manifold with both continuous product and inverse operations, and only after that comes the Lie group, which is a differentiable manifold with differentiable product and inverse operations.
3) similarly, the Riemannian manifolds are a subset of the differentiable ones.
So whenever I am exposed to formal mathematics, there are always all these definitions of things that we normally take for granted, and I get confused by how it could be otherwise. So with the Hausdorff space . . . what is an example of a space that is not a Hausdorff space? How could distinct pooints not have distinct neighborhoods? What does it mean for the points to be distinct, then?
Also, you can tie into categories again. A topological group is a group object in Top, and a Lie group is a group object in Man.
ME: consider a copy of the line. Now at some place along it, add another point. It looks something like
—-:—-
You could remove either of those two points and be left with a line. Any open neighborhood of either point contains a bit of the line to the left and a bit to the right, so they have no disjoint neighborhoods.
In practice, though, property 3 is the important one. Properties 1 and 2 are just there to exclude certain pathological cases which we don’t want to have bothering us later. The countable basis thing is there to get rid of the long line.
Your definition of Hausdorff is really the definition of T_1
(points are closed). Hausdorff says that distinct points have disjoint neighborhoods.
The previous post gives an example of a space which is T_1 but not Hausdorff.
Not quite, as I’m sure you’re aware. Actually a Riemannian manifold is a differentiable (a.k.a. smooth) manifold with some extra structure which defines length and angle locally; this structure is called a metric. Every smooth manifold has many metrics. In fact, proving this and some related facts is much of the reason for condition 1 above.
In category-theoretic terms, our category of Riemannian manifolds (call it Riem) is not a subcategory of Man; rather, it is a separate category, with a obvious forgetful functor F: Riem -> Man. Moreover, the object function of this functor is onto, since every manifold has at least one metric.
Chad, you’re right of course.
Okay.. so there’s a couple things I’m confused about.
One is “basis”. This does not seem entirely similar to the definition of “basis” I am familiar with. What would be the basis in this sense for the real line, or the 2D plane? It also seems a bit confusing to me how that definition of a basis could even be used in practice– what happens if there are sets in the topology which *cannot* be constructed by a finite number of intersections and unions? Say, maybe the Cantor set, whose definition involves an infinite number of intersections. Are such sets somehow technically not in the “topology”, or is the trick that no such will ever wind up being open, or what?
And what, exactly, IS a lie group? I’m looking at the wikipedia definition– would it be accurate to say a lie group is just a group whose member set is a manifold and whose group operation is a continuous function? Is (R, +) a lie group?
Open balls form the basis for any Euclidean space.
For example, basis open sets on the real line are just open intervals (a,b). Basis open sets in the plane are open disks centered at any point (x,y) with any radius r.
Also, the sets that are “in the topology” are the ones we’re going to call “open.” So the Cantor set not being in the topology means it’s not open.
Oops, time to teach, or I’d post more!
Coin:
The basis of a topology isn’t a *unique* set of open sets. *Any* set of sets which can form any set in the topology by a finite sequence of unions and intersections is *a* basis.
For an example of why a basis matters: there are plenty of non-countable sets. The structure that you want from a manifold doesn’t work if there isn’t a countable basis. It’s actually a very interesting idea, which I’ll write more about in a future post: but a manifold is 2-countable, which means that the set of points in the manifold may not be countable, but there is a basis with a countable (if infinite) number of sets.
So.. Think about the real number line. It doesn’t contain a countable number of points. And as a topology, it doesn’t contain a countable number of sets. (If it *did* have a countable number of sets, then it wouldn’t be Hausdorff: a countable number of sets with an uncountable number of points would result in being unable to define neighborhoods to separate each point from all of the others.)
But you *can* define a countable basis for the topology. Taking open balls of every rational number size around every rational-number on the line gives you a basis for the set which is countable.
Coin:
For your second question: yes, a lie group is a group whose value set is a manifold, and who’s group operator is a continuous function in the topology.
Chad says : “Moreover, the object function of this functor is onto, since every manifold has at least one metric.”
Actually, not quite. There is an example given on this page of a non-metric manifold of dimension 1.
This was mentioned before, but you must allow arbitrary unions to be formed to recover the entire topology from a basis.
Example: the usual real line R (-&infty;, &infty)
Basis: B, the collection of open intervals of finite extent, all (a,b) where a
David:
The long line, you mean? You’re right, it doesn’t support a metric. However, I don’t consider this a manifold; as I understand it, most mathematicians don’t, for essentially this reason. Note the reference to “condition 1” in my first comment.
*Ahem* — not all non-Hausdorff topologies are pathological. My field considers the Zariski topology to be the “right” one, and that’s highly non-Hausdorff. 🙂
Lie group products have to be not just continuous, but also differentiable, if I’m not mistaken.
Really, the easiest way to think of Lie groups is to look at groups of matrices – GL(n, C) and its closed subgroups. That won’t get you all Lie groups you want, but it’s close enough in a certain sense.
Davis, the Zariski topology isn’t Hausdorff, but it doesn’t satisfy property 3), either. The examples you need to exclude with properties 1) and 2) are pretty pathological, in the sense that they’re useful mostly when studying counterexamples; the long line and John Armstrong’s example of a non-Hausdorff space are prime examples of that.
As we are discussing pathologies, is it the requirement for finite composition from bases that restricts n less than oo here? IIRC there is a similar restriction on measures.
I don’t see how to construct what happens, but Mathworld hints that several topologies becomes possible for infinite-dimensional spaces. ( http://mathworld.wolfram.com/BanachSpace.html ) So something becomes qualitatively different. (Well, perhaps the n-ball volume goes to oo too, but I don’t see how that connects to topology.)
“several topologies becomes possible”
Umm, doesn’t sound right, several topologies should be possible anyway. And in the example, it rather seems topologies gets different behaviour.
This comment is attempt to demonstrate how over 500 million years of evolution [natural selection?] may use string theory [or loop] topologies.
This comment addresses cellular cooperation within an organism with respect to competition between organisms for ‘Auditory Space’.
Consider an article from a special edition of Scientific American ‘Secrets of the Senses’ 12 DEC 2006 by Masakazu Konishi [CIT] “Listening with two ears” with page 30 ‘Coordinates of Sound’ diagrams.
From my perspective this ‘Auditory Space’ may be a stereoscopic dual coordinate topology possibly employing a vector operator algebra similar to that of Borcherds for a 3D-space with string and time dimensions. In this example, the string dimension may be the sound energy dimension referred to as intensity by the author. Intensity may be equivalent to the amplitude of the energy string vibration or frequency. The author does discuss possible neural algorithms.
A CIT press release discusses [without diagrams] this work of Konishi and Eric Knudsen awarded the Peter Gruber Foundation Neuroscience Prize in JUL 2005.
http://pr.caltech.edu/media/Press_Releases/PR12719.html
What?
Hi Coin:
1 – Your What? follows my comment so:
Richard Ewen Borcherds in 1992 [awarded a Fields Medal 1998] proved the ‘Monstrous Moonshine’ which conjectured the unexpected ” … connection between the monster group M and modular functions (particularly, the j function)”.
http://en.wikipedia.org/wiki/Monstrous_moonshine
Borcherds developed vertex operator algebra [VOA] which took the physics concepts of strings and made this a string dimension in his mathematics.
http://en.wikipedia.org/wiki/Vertex_operator_algebra
2 – The important page 30 ‘Coordinates of Sound’ diagram, I have only seen in the SCIAM article previously referenced [pages 28-35].
Some related concepts and diagrams are on the web:
‘Auditory Cues and Ecolocation; Psychology 403, Section Q1 Animal Navigation and Wayfinding’ [Michael Snyder, U-Alberta CA]. http://www.psych.ualberta.ca/~msnyder/Academic/psych403/week8/w8oh.html
My perspective is of course a modified and speculative use of VOA.
No rigorous proof of actual topologies have as yet been given – Nobel prizes have not been awarded.
That press release is significantly easier to understand. (Having studied a bit of that work for a computational neuroscience class, I can testify that it’s pretty darn cool. Even cooler than most things which involve sticking electrodes into living brains. Brains. Braa-ai-ai-ai-nnnnssss. . . Ahem. Sorry.) The comment which references that press release, however, I can’t figure out.
Davis: In algebraic geometry the Zariski topology is very useful, but in differential geometry having a non-closed diagonal is pathological and throws off all sorts of proofs. “Pathology” depends on context. If the post were about varieties, you’d be entirely correct that Zariski is “right”.
My field considers the Zariski topology to be the “right” one, and that’s highly non-Hausdorff.
I was always told that the Zariski topology is evil. The open sets are too big. Now Grothedieck topologies — those are cool.
At this point, I should probably mumble something like fpqc being “right”er or something like that, but I’m already getting a bit out of my depth.
Torbjörn, it’s not that hard to show that you can impose many inequivalent topologies on infinite-dimensional spaces. I’m not sure how you show that the L^p metrics are inequivalent for p1 != p2, though I know a proof for showing that L^2 is inequivalent to the rest of the L^p’s. But it’s not that hard to come up with a separable space and an inseparable one with the same Hamel basis cardinality.
Aaron Bergman: Grothendieck topologies are indeed cool. They’re ways of putting topologies on arbitrary topoi, which I’m sure our host will talk about sometime since they’re all sorts of related to categorical approaches to logic.
Alon, thanks, that was really helpful! Perhaps I should have thought about L^p norms myself, I’m somewhat familiar with the concept, and what you say seems right. L^2 functions are AFAIK analogous to (energy, probability) densities, so they should be nicely defined and behaved.
The problem is that I still failed to see the difference between n being finite and infinite for the space. But from what you said I remembered an L^1/L^2 discussion in distribution theory, and that n infinite is studied for vector spaces of functions. Friedlander “Introduction to distributions” pp 90-127 discusses the Fourier transform. A difference between L^1 and L^2 is that the Fourier transform is not a map Cc oo ->Cc oo for L^1 test functions by analytic properties. (So one use seminorms to be complete.) But in L^2 that isn’t a problem.
And by that route I found http://mathworld.wolfram.com/BanachSpace.html which simply discuss why norms are similar for n finite but becomes different for n infinite by comparing the sup norm with the L^1 norm on patological examples, the first is found to be complete, the second again not.
“for L^1 test functions” – for L^1 and test functions