# Sets with Structure

By Bobby Neal Winters
This
semester I am teaching a course in topology after a hiatus of six
years. I am using a classic text by James R. Munkres with the title,
fitting enough, Topology.
This is the text I had my graduate level course from. It’s been like
meeting an old friend again after an extended separation. Only someone
who’s done that can appreciate all of the levels of meaning.

Topology
is a word-like every word now that I think of it--that carries a bundle
of meanings. On the level that is most accessible to a popular
audience, it is understood to mean that branch of mathematics in which a
coffee cup is no different than a phonograph record. (That’s a CD to
you, you young whipper-snappers!) For the sake of precision, we could
make a distinction by saying geometric topology or even low-dimensional
topology, but in practice clarifying adjectives or adjectival phrases
get stripped off and we are left with topology left alone, forced to
hide the other meanings it carries.

Today,
I would like to venture into one of those areas where angels fear to
tread to talk about the subject that mathematicians (especially
geometric and low-dimensional topologists) refer to as general or
point-set topology.

One
can could say that low-dimensional topology is a sub-speciality of
general topology, and I will justify the sense in which that is true in
the sequel, but such a statement blurs over differences of mindset among
the various practitioners.

Let
me say I was drawn to my first topology course having seen the pictures
of coffee cups being blithely changed into phonograph records, donuts,
etc, only to find something entirely different.

A
course in general topology begins with a topological space. A
topological space is about as abstract a concept as the math major will
meet as an advance undergraduate or beginning graduate student. It is a
set which is paired with a special collection of its own subsets, and
this special collection of subsets have a set of laws they must obey. I
won’t tell you now no matter how much you beg me. We give a name to
that special collection of subsets and call it a topology. I told you
the word carried a bundle of meanings.

The
most common example of a topological space is the set of real numbers.
Topologists who’ve just read that sentence are now picking up pencils
from their desks to write in “with the usual topology” between the “s”
in the word numbers and the period that follows it. I left it out on
purpose just to annoy them because it is the usual topology. It is based
on the open intervals that students learn about as early as middle
school. The open intervals are used to construct open sets and the set
of all of the open sets of the real numbers is the usual topology on the
real numbers.

The
usual topology on the real numbers is such a natural thing to us--and
my “us” I mean “geeky math types”--we don’t even notice that it’s there.
We use the real numbers with the usual topology first in calculus and
later in analysis, and I have talk these courses without ever uttering
the word topology. Most of the basic results in those areas can be
reached without naming the topological concepts explicitly.

Perhaps
the concept of a topological space would never have been created had
mathematicians not ventured beyond the real numbers, but--you know those
scamps--they did. They ventured into the plane, into 3-space, into
sets of functions, and so forth, and they discovered sets of subsets in
each of those areas that behaved like the open subsets of the real
numbers behaved.

If
I knew more of the history of the subject, this would be an opportunity
to segue into a case study in abstraction. Those three examples I
listed above have quite a bit of structure on them. They have ways of
doing arithmetic, they have ways of measuring angles, and they have ways
of measuring distance. They are groups; they are vector spaces.

When
we push out to the level of abstraction required by the topological
space, we forget about all of that other structure. You can’t do
arithmetic; you can’t measure distances. You think about only the set
and its topology. You only define properties that can be discussed in
terms of the members of the topology. You only discuss functions which
respect the members of the topology.

In
some sense, learning general topology first requires that you forget
everything else you know about anything. You become a slow thinker; you
become a deliberate thinker; you always must be careful that your
intuition--raised as it was in the fertile fields (nerdy pun fully
intended) of the real numbers--does not lead you astray.

This
sort of abstraction allows us to prove theorems that apply to a wide
range of areas. It allows us to create language to see an underlying
unity in diverse areas of knowledge. It also provides a trap-door into
what has been referred to as centipede mathematics, as in “How many legs
can I pull off the centipede before it can’t walk any more?”

I
called it a trap door, but I am not sure that metaphor works. It makes
what happens sound like an accident. The truth is more complex.
Many--most--who are drawn into mathematics find this sort of
abstraction attractive, not to say intoxicating. Going deeper and
deeper into abstraction leads us into what our appetite desires. It is
like the wind buoying up our wings, lifting us farther and farther from
the ground. Here the story of Icarus is attractive, but also
inaccurate. We don’t go so high that the sun melts our wings; we are
lifted so high we are never seen again.

There
is a quote I’ve heard attributed to RH Bing, a Texas mathematician who
is a personal hero of mine. When asked about a visiting topologist, he
is said to have replied, “He studies spaces of which there are only one
example and only in England.”

Mathematics,
especially abstract mathematics, is best when it is equipped with
numerous examples. Examples give breadth and richness. Examples
guarantee you aren’t just proving theorems about the empty set. But I
digress.

General topology is alive with examples. It is wide and it is deep.

There
was a time in my career, and I will say this without shame, that I
taught subjects simply because I wanted to learn them myself, without
regard to the student. I say it without shame because the students can
still get a lot of value from that provided they are motivated
themselves and their needs are being regarded other places. Time has
dealt with me in any case. I find myself singing along with Bob Seger:

Well those drifter's days are past me now

I've got so much more to think about

Deadlines and commitments

What to leave in, what to leave out

As
I teach my courses now, I try to focus on what I think the student
needs. One great need that students have as they enter into graduate
mathematics is to have their pre-assumptions stripped away. The
abstractness of general topology is the best method I know. That having
been said, there is so much of it. What do I leave in? What do I leave
out?

In
the end, my prejudice is to choose topics that will lead my students
toward areas where mathematics is growing, places where many branches
come together, places where there is structure--much structure. Then
they will be able to choose.

## 1 comment:

:) Taking away preconceived notions definitely happened in my first topology class, but I learned so much about things I thought I understood!

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