From: mathwft@math.canterbury.ac.nz (Bill Taylor) Subject: Re: What is topology Date: 25 May 1999 10:26:59 GMT Newsgroups: sci.math Jim Ferry writes: |> I felt the same way about my Topology course. We proved a bunch |> of stuff, but why? Even now I don't know much about topology, |> but maybe I can mumble something useful. Put it there, kid!! My life exactly! :) Looking back at my final year's notes, I can see it was reasonably well motivated, in that we started with Euclidean, then general metric spaces, and finally general topologies; but even so it still seemed like nothing but a bunch of boring theorems... Looks like there were once a whole lot of bad topology instructors around. Hopefully they're better now. |> set function bijection |> group homomorphism isomorphism Yes. Though maybe there's the opposite problem here... lot's of motivation (to put many parts of math on similar footing; to provide a uniform language; to identify significant concepts); but seemingly very few theorems! I dimly recall 9's lemmas and similar things, but none of it seemed to be terribly significant. Just showing my ignorance here, of course, but hey - what else is new!? I can imagine John Baez spinning in his office chair... BTW, I loved the phrase and image of whoever (forgotten, sorry) it was said categories were like "sets with attitude"... marbles like fighting ferrets! Someone else made the point that Euler's "v-e+f=2" was a good example of topology application. And so it is. But it leads me to a thought I've often thunk before. There seem to be two almost disjoint types of topology, discrete and continuous. The latter, involving continua, connectedness, continuity, compactness, convergence, and other con-phenomena, is much more like analysis, analysis situs in fact. Whereas the former, involving glueing manifolds, orientability, homology groups, coloring and other partition phenomenon, is almost algebra; it might be called combinatorial topology. They are almost two disjoint subjects. Yes, they are intimately connected at the BOTTOM, where the latter can only be properly defined by reference to the former, but still, after development, they are almost independent. A situation not unlike the ultimate near-independence of geometric optics from its basis in QED; and many other analogems as well. OK; not a very profound observation maybe, but it strikes me that way. -------------------------------------------------------------------------- Bill Taylor W.Taylor@math.canterbury.ac.nz -------------------------------------------------------------------------- Psychologists and sociologists only mark time noisily, while molecular biologists work out the implications of the DNA molecule. -------------------------------------------------------------------------- ============================================================================== From: rusin@vesuvius.math.niu.edu (Dave Rusin) Subject: Re: What is topology Date: 28 May 1999 15:15:03 GMT Newsgroups: sci.math Bill Taylor wrote: >There seem to be two almost disjoint types of topology, discrete and >continuous. The latter, involving continua, connectedness, continuity, >compactness, convergence, and other con-phenomena, is much more like >analysis, analysis situs in fact. Whereas the former, involving glueing >manifolds, orientability, homology groups, coloring and other partition >phenomenon, is almost algebra; it might be called combinatorial topology. >They are almost two disjoint subjects. Yes, they are intimately connected >at the BOTTOM, where the latter can only be properly defined by reference >to the former, but still, after development, they are almost independent. This is a good point. Since I am teaching "Introduction to Topology" next term, I ought to be able to amplify this :-) Part of the problem here is that we naturally assume that everything called "topology" is closely related and yet is clearly distinct from everything not called "topology". Given the evolution of mathematics and the connectedness (ahem) of its branches, I don't know if that's really a reasonable assumption any more. I know topologists who typically publish in, say, Journal of Pure and Applied Algebra, and who can scarcely read papers in the journal Topology and its Applications. So it may be more productive to consider the different historical bases for topology, and the other fields with which parts of topology interact. One way to think about topology is as an outgrowth of Set Theory. As we see repeatedly in this newsgroup, there is a natural aversion to the complete flexibility afforded by the typical axioms of Set Theory, e.g. the Banach-Tarski paradox. One possible response to this is to say, "OK, but I'm not interested in just any old set, I want 'normal-looking' sets and functions". These are hard concepts to pin down, but one reasonable place to start is with open or closed intervals, and continuous functions. So what Topology "is" is an intermediate field of study between Set Theory and, say, Analysis. Going in the opposite direction, it is this connection with Analysis which has probably kept Topology in the curriculum at many schools. As others have mentioned in this thread, Topology is a flexible arena in which one may study convergence and that kind of thing. For example, one must study some Functional Analysis in order to make headway in Differential Equations; Functional Analysis is (more or less) the study of vector spaces of functions (e.g. the integrable functions, the C-infinity ones, etc.) This is "just" linear algebra except that all the good vector spaces are infinite-dimensional, and so one has to be clear what terms like "basis" and "linear combination" really mean. This leads to questions of convergence, and thus, of topology. (One may argue that Functional Analysis is the study of _topological_ vector spaces). Another poster mentioned "analysis situs"; this is another reason an analyst might consider Topology to be "natural". Consider the question of whether line integrals are path-independent (from a physical point of view, this asks whether a force field against which you must work is a conservative force -- does the amount of work done depend on the path, or only on the endpoints?). A typical proof of path independence assumes a vector field (P,Q) defined on a star-shaped domain in the plane, and proves that path integrals are path-independent in the expected cases. Yet this "star-shaped" condition is really misleading; it turns out that exactly what's needed is that the domain have no "holes". Of course, deciding just what a "hole" is turns out to be a little tricky, but that's exactly what Topology is for: it gives the language necessary for describing the (global) constraints on analytical problems. I guess the general sweep of topics between Set Theory and Analysis is called "General Topology" or "Point-Set Topology". Another part of Topology is usually described as having arisen from the Koeningsberg Bridges problem. This is really a topic in Graph Theory, nowadays, but Graph Theory is "just" the study of 1-dimensional CW complexes anyway :-) The key idea here is that we have a shape built up of simple parts and we are interested in questions of how those parts are connected. Perhaps you can see the link with the topic of "holes" mentioned above. I take it this is the sense in which the word "topology" is used by those who discuss the architecture of computer networks. A related topic has also been mentioned in this thread: Euler's formula v-e+f=2 relating the number of vertices, edges, and faces in convex polyhedra. Now there are three basic parts being connected, instead of just the two in graphs. As you might imagine, the process continues in higher dimensions: one looks to see what basic information about an object can be discerned from its parts. In some sense this idea takes you in the wrong direction: you might think from this description that topology can be very combinatorial. I suppose that's true in some areas, but most people would say that the particular combinatorial makeup of an object is not as relevant as the overall shape. A triangle which is split into two smaller triangles is a different combinatorial object, but really "the same" as before. So instead of combinatorics, you should be looking at this topic as asking for simple concrete data obtained from the object which won't change under such trivial modifications. Is it connected? Does it have holes? etc. These questions turn into numerical questions (How many components?), which historically changed to algebraic ones (e.g. the Betti numbers became the numbers of generators of certain groups attached to the spaces). This broad section of Topology is now "Algebraic Topology", and has spun off related disciplines of Category Theory and I suppose K-Theory. There has been a lot of successful cross-fertilization with parts of algebra, notably Ring Theory and Group Theory. In practice, Algebraic Topology seems primarily concerned with the algebraic invariants rather than the underlying shapes; those will be taken up again in the next section. Next is the "coffee-cup" school of thought about Topology. Here the natural place of origin is Geometry (whatever that is!). Consider Euler's Formula again. You may have noticed I mentioned "convex polyhedra". In another thread someone recently mentioned this formula without restricting the polyhedra. Well, in the general setting, Euler's formula is wrong: try counting vertices, edges, and faces in the Pentagon [US military headquarter]; you'll find v-e+f = 0. On the other hand, convexity isn't really the right condition needed here: clearly if one vertex of an icosahedron pointed "in" instead of "out", the numbers v,e, and f are unchanged but convexity is lost. So what is the right condition here? It turns out that v-e+f=2 is valid for any polyhedron which has the same underlying topological structure as the sphere. Now, this is really the same point I've made when discussing Algebraic Topology, and it's also really the "no-holes"-vs-"star-shaped" problem I mentioned earlier. But now I want to focus on honest-to-goodness geometric objects. In topology we discuss the Moebius strip without specifying just how rapidly it twists around itself, we specify "coffee cup" without discussing capacity or diameter, and so on: these are real geometric objects, and yet we focus on the underlying shape, free to make smooth distortions. I suppose this is called "Geometric Topology", and it includes all the parts of topology which are used to entice unsuspecting schoolchildren to investigate the subject: those topological puzzles with chains and loops, Klein bottles, Knot Theory, and so on. In general the _tools_ used here are the same as the ones in Point-Set and Algebraic Topology, but the _goal_ is a perhaps little clearer. Finally, I might mention "Differential Topology", which is closely related to the geometric side, since most of the geometric shapes we play with are "nice" in the sense that they are curves or surfaces: every point has a neighborhood which looks just like the line or the plane. These objects (and their higher-dimensional analogues) are _manifolds_, and they have the feature that, since they're locally the same as Euclidean space anyway, you can do with them any local things you do in Euclidean space. A couple of such tricks I might mention are to solve differential equations, and to measure distances. Well, there are some additional constructions you'd need to put into place in order to make these ideas work, but once you do so you have a very fallow area of inquiry. For example, meteorology can be described with differential equations on the surface of a sphere; general relativity is a study of distortions of the metric on (what we imagine to be) a manifold. Traditionally the study of metric geometry on manifolds is a separate topic (Differential Geometry) while Differential Topology looks to see if stronger topological statements can be made with the extra hypothesis that the spaces are so nice. Quite a bit of this discussion must sound rather vague, with "so" many "words" in "'quotes'". This is perhaps why some students find it hard to be comfortable with topology: you think you know what the torus is, and then when you have to prove something about it, you find you don't really have a good language to match your intuition. Of course this can all be placed on a solid footing, but if the definitions are deferred until after the discussion, the topic sounds unfounded; if the definitions are given first and then the fun examples, the topic sounds deadly dull. The different parts of Topology are given different classifications in the Mathematics Subject Classification. For further information, visit General Topology: index/54-XX.html Algebraic Topology: index/55-XX.html Geometric Topology: index/55-XX.html dave ============================================================================== From: "Dr. Michael Albert" Subject: Re: What is topology Date: Sun, 30 May 1999 19:35:15 -0400 Newsgroups: sci.math First, let me say that I really enjoyed Dave Rusin's post. Now, to the student who started this thread, I suspect the student took a course which emphasized "point set toplogy." I assume that the course made everyone aware of the fact that all subsets of Euclidean and all metric spaces are indeed topologies, and I think this is good pedagogy. I think it's also important, however, to quickly introduce students to "natural" examples of topologies which are not metric spaces in order to both convince the student that the extra generality is potentially useful and to make sure the student doesn't start assuming all of the usual metric space properties. (Another useful thing is to show that all metrics on finite dimensional vector spaces induce the same topology, which shows that topology is catching some "abstraction" of the metric properties, but is also discarding some details of the metric structure). I think the most useful example is to discuss pointwise convergence of functions and the "[weak] product topology". It is "natural" to want to talk about convergence of sequences of functions, and in turn "open sets" of functions, etc, but there is no useful "metric" which corresponds to pointwise convergence. Another good exmaple is to introduce on the real line the topology whose base is the set of all intervals of the form {x:x Subject: Re: What is topology Date: Sat, 22 May 1999 10:31:37 -0400 Newsgroups: sci.math In article <7i4ra7$maa@news.acns.nwu.edu>, Miguel A. Lerma wrote: > When I studied Topology for the first time I arrived to the > conclusion that the concept of "topological space" is the most > general one in which the concepts of "limit" and "continuity" > make sense. There are more general ones, but topological space is the most general one that is commonly used. There is a giant text by Cech (Topological Spaces, Prague, 1966) where practically everything is done in a more general "closure space" or "pretopology" context. Another more general one is the "pseudotopology". [My paper on this is: Three cryptoisomorphism theorems. Studies in foundations and combinatorics, pp. 49--60, Adv. in Math. Suppl. Stud., 1, Academic Press, New York-London, 1978.] -- Gerald A. Edgar edgar@math.ohio-state.edu Department of Mathematics telephone: 614-292-0395 (Office) The Ohio State University 614-292-4975 (Math. Dept.) Columbus, OH 43210 614-292-1479 (Dept. Fax)