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.
--------------------------------------------------------------------------
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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:
http://www.math-atlas.org/index/54-XX.html
Algebraic Topology:
http://www.math-atlas.org/index/55-XX.html
Geometric Topology:
http://www.math-atlas.org/index/55-XX.html
dave
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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)
**