Notions of Sameness

Notions of Sameness

By Bobby Neal Winters

Mathematicians use sets and structures on sets to model realities that they perceive only in their minds.  I dealt with this in the matter of topology in one of my recent essays.  These structures induce special properties that we want to preserve.  I am still interested in this mainly in the case of topological spaces, but I will work my way up to it via a series of examples from other areas of mathematics.

The first example is that of sets with no structure in which we can only discuss the concept of cardinality.  This is to say, how many elements does a set contain?

With finite sets we can say that a set is composed of five elements or seventeen elements or two trillion five hundred fifty four billion two hundred twenty seven million three hundred twelve thousand five hundred and two elements.  There is a number we can associate with them.  

For infinite sets, the above does not work; we have to do a flanking maneuver. We avoid the word number and use the term cardinality.  We say that two sets have the save cardinality if they can be put into one-to-one correspondence with each other.  

We use the language of functions to make such notions precise.  Functions are thought of as having a sending set (the domain) and a receiving set (the range).  The range is like a target.  When two sets have the same cardinality, there is a function from one set to the other that is one-to-one and onto.  A function is onto if every point in the receiving set is matched with a point in the sending set.  A function is one-to-one if no set in the receiving set is matched twice.

A function that has both of these properties at once is christened with the high-falutin’ label of bijection.  It’s technical and pretentious but remember it anyway because it’s important.

If two sets have a bijection between them, they have the same cardinality.  If one (and therefore the other) of those sets is finite, they will also have the same number of points.  By this mathematical slight of hand of using a particular type of function to create a new notion of “sameness” we’ve expanded the notion of number/size into the notion cardinality/size.

This is done in the realm of algebra as well.  And here I must be careful because I am using the word “algebra” differently than many use it.  If you had pre-algebra and algebra in middle and/or high school, or if you have had intermediate algebra or college algebra in college, then rest assured that I am talking about something that is related to that in the way that a Tomahawk Missile is related to, well, a tomahawk.

Algebraists, and I am over-simplifying, study sets with binary operations on them.  For example, they might study the real numbers with the operation of addition or the positive real numbers with the operation of multiplication.  (Notice I put the word “might” in there because they aren’t really interested in those two particular sets, but I will use them because I don’t want to talk about a lot of algebra now.)

The real numbers numbers with addition and the positive real numbers with multiplication are examples of groups.  Groups are sets with binary operations that obey certain rules.  Don’t worry about what those rules are; if you can’t help it, look it up on Wikipedia.

Those of you who have had a course in college algebra might remember the logarithm.  I put emphasis on the word might there because I’ve no doubt that you have seen the logarithm.  You might remember it; you might wake up in cold sweat screaming it at the top of your lungs; your lover might have attempted to comfort you afterwards asking, “What is a logarithm, Sweet heart?” only to have you deny ever having heard the word.

In any case, the logarithm is a function from the positive real numbers to the real numbers that is bijective, i.e. it is one of those important one-to-one correspondences.  The important thing about it is that it changes the operation of times to that of plus: log(a times b) is equal to log(a) plus log (b). In mathematical ages B.C.--before calculators--the logarithm was used as a means of doing multiplication by turning it into addition.  You may not have notice, but addition is a lot easier than multiplication.  

Such a function that is bijective and preserves the operations on the groups--in this case turning multiplication into addition--is called an isomorphism.  The groups are said to be isomorphic to each other and a mathematician in the capacity of algebraist can’t tell the difference between them.

I could go on listing the various areas of mathematics and the functions they use to preserve there important concepts, but instead of doing that let me focus on topology.

A topological space is a set along with certain special subsets that we call “open” sets.  Topological concepts are those concepts that can be defined in terms of open sets.  As a consequence of this, we are interested in functions that preserve open sets.  A bijective function between two topological spaces that takes an open set in the first to an open set in the second is called a homeomorphism.  A pair of spaces between which there is a homeomorphism are said to be homeomorphic.  Spaces that are homeomorphic are the same as far as a topologist is concerned.  

Most frequently, I see homeomorphisms defined differently.  We usually define a homeomorphism to be a continuous function which is invertible and whose inverse is continuous.  This is mathematically equivalent to the other definition I’ve just given, but it’s shorter.  

Continuous functions are interesting in their own right.  Indeed, one might argue that topology was created in order that we might better study continuous functions.  There is also quite a well-developed theory of continuous functions that allows us to check whether a particular function is continuous.  For example, it is quite easy to use calculus to check that the logarithm and its inverse the exponential function are both continuous; it might be less easy to show they preserve open sets.

But in our desire to be efficient with our definitions, we should take care not to obscure for the beginner the very thing that homeomorphisms do: they preserve open sets.  As they preserve, nay set up a one-to-one correspondence between the open sets of each space, any properties defined in terms of open sets will be preserved.  That is the point.

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.