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.