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.