From: lrudolph@panix.com (Lee Rudolph) Subject: Re: "Topologies" Date: 1 Aug 2000 15:08:48 -0400 Newsgroups: sci.math Summary: [missing] Jeremy Boden writes: >I'm planning on taking an Open University (undergraduate) course next >year entitled "Metric and Topological Spaces and Geometric Topology" > >Since the course description ...by which I assume you refer to the one posted at http://www-icdl.open.ac.uk/icdl/export/europe/unitedki/openuniv/cour/m435.htm >is somewhat brief, I'd be grateful if >someone could a *short* description of what the relationship is of the >two subject areas (if any). Well, I can try, with an eye on the course description. As it says, the part of the course on "metric and topological spaces" is an introduction to (the category T of) topological spaces and continuous mappings between them, with (presumably) a lot of emphasis on the special class of examples given by metric spaces and continuous maps. Now, T is a big category, filled with all sorts of behavior that some people think is just peachy-keen and other people label "pathology". There's even a lot of "pathology" among metric spaces. One school of pedagogy thinks that students will do better if they're exposed early to lots of "pathology" (and, of course, led to reason carefully about it), while another thinks that it's better for beginners to have their first experiences carefully limited to "non-pathological", "well- behaved" topological spaces (again, of course, being led to reason carefully in this more restricted context). There are various possible interpretations of "well-behaved", but certainly almost everyone agrees that Euclidian spaces--in particular, the real line and the real plane (AKA the complex line)--are well-behaved. Spaces that are "locally like" the real line or the real plane (in a sense that can be nicely described once the student is familiar with T) are then probably the next most "well-behaved" spaces. These include the circle (locally like the line) and the various surfaces, including "the sphere, the torus, the Mobius band and the Klein bottle" as mentioned in the course-description. So, we've got some spaces, but what do we do with them? There is, as a matter of fact, plenty of "point-set topology" (the systematic investigation of pathology...oops, I've just betrayed my bias) available in the world of surfaces. But there's also room to really get somewhere by "geometric" (as, vaguely, contrasted with "set-theoretical" or "axiomatic") methods, largely because the local niceness of the surfaces allows one to (locally) use some of the geometric machinery that's already in place in the real plane. And it turns out that, not only is there room to get somewhere, but there's somewhere (intersting) to get to. That is: as I said, T is a big category; one way to study it is to try to find meaningful and calculable "functors" from T (or various subcategories U, V,... or quotient categories Q, R, ..., or subquotient categories G, H,...) into smaller, more manageable categories. One goal of (some people's version(s) of) topology is to find enough different functors that an object in your topological category C (that is, a topological space, subject to possible restrictions, and possibly only specifiefd up to some equivalence relation) is essentially known completely as soon as the values of all those functors on that object are known. Well, this goal is usually inaccessible in that generality. But one place where it is accessible is in the world (I hesitate to say category any more) of surfaces. In fact, there is a complete "combinatorial classification" of surfaces. And it can be done using very geometric techniques (though it can also be done using very ungeometric, or at least not transparently geometric, techniques as well). I see I haven't been brief. And I guess I haven't really answered what the relation between the two half-subjects of the course is, either. Okay: you can't do the second half rigorously without the first half. On the other hand, having done the first half you could go off in many another direction besides the (given) second half. Lee Rudolph