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What is Math 269 About?

This course is a good follow-up after Math 240 (but that course is not a prerequisite). Math 269 has a substantial overlap with Math 276 (i.e., a metric space is an important special type of topological space), so students probably should not take both 269 and 276. In particular, math graduate students should not take 269. (In that respect, the catalog description is incorrect.)

What is this course about? Here is a very brief overview. It's really about two subjects, set theory and metric spaces; each will occupy about half the semester. By the way, there is some overlap between this course and some other courses taught by our department.

Set theory was started by Georg Cantor, about 100 years ago. You might say that he "tamed" infinity, taking it away from the theologians and making it a proper part of mathematics, with its own methods of calculation.

Centuries earlier, mathematicians had noted that there exists a one-to-one correspondence between the two sets {1,2,3,...} and {2,4,6,...}. Thus, in some sense, an infinite set is the same "size" as a proper subset of itself. This puzzled earlier mathematicians, but Cantor realized that this is not a contradiction at all. We say that two sets have the same cardinality if there exists a one-to-one correspondence between the two sets. Cantor proved that the integers and the rationals have the same cardinality, but the real numbers have strictly greater cardinality. (Thus, there are at least two different "sizes" of infinite sets.) Moreover, if X is any set, then the collection of all subsets of X has strictly greater cardinality than X. (Consequently, there are infinitely many different sizes of infinite sets!) In Math 269 we will study the "arithmetic" of cardinality.

For finite sets, cardinality and ordinality are the same thing; but for infinite sets, cardinals and ordinals are different. Ordinals are sets for ordering -- i.e., for forming concepts like "first", "second", "third", etc. A well-ordered set is a set equipped with an ordering, such that each nonempty subset has a lowest member. Examples are

{0,1,2,3,...,99}, with its usual ordering
{0,1,2,3,...}, with its usual ordering; this is the first infinite ordinal.
{0,1,2,3,...,w}, where w is a "number" larger than all the integers; this is the second infinite ordinal.
{0,1,2,3,...,w,w+1} -- the third infinite ordinal.

Note that the first three infinite ordinals all have the same cardinality. In Math 269 we will also study the "arithmetic" of ordinals, though we may devote less time to this topic than to cardinals.

The catalog description of this course mentions, among other things, the Axiom of Choice (AC). That's a rather glamorous subject; it's the most well-known nonconstructive existence principle in mathematics. It's a favorite subject of mine -- in fact, I've even created a whole web page about AC. AC has many interesting reformulations and applications in topology, algebra, functional analysis, and other branches of math. A few of them are even related to set theory -- for instance, the statement that

If X and Y are any two sets, then one of them has
the same cardinality as some subset of the other

is equivalent to the Axiom of Choice. However, I might not spend much time on AC in Math 269, because relatively few of its applications are accessible without more prerequisite material than that supplied by Math 269.

Metric spaces were invented by Frechet about 100 years ago. The concept is just an abstract version of the notion of distance. Different metrics are useful for different purposes. For instance, you already know that between two points (x₁,y₁) and (x₂,y₂) in the plane, the distance "as the crow flies" is

d(x,y) = [(x₁-x₂)² +(y₁-y₂)²]^1/2. But the distance "as the taxicab drives in Manhattan" would be measured more usefully as d(x,y) = |x₁-x₂| +|y₁-y₂|, because that's the distance the taxi must actually travel -- since the taxi driver is not allowed to cut through people's backyards or the first floors of buildings.

A metric on a set X is a function d satisfying these axioms: for all points x,y,z in X,

d(x,y) is a number, with 0 < d(x,y) < infinity.
d(x,y) = 0 if and only if x = y.
d(x,y) = d(y,x).
d(x,y) < d(x,z) + d(z,y).

In Math 269 we'll spend an hour or two considering various different examples of metric spaces. But most of our study of metric spaces will not be concerned with those particular examples. Rather, it will be concerned with the abstract theory of metric spaces -- i.e., with properties and concepts that make sense just using the axioms, without regard to which particular metric we're working with. When does a sequence converge to a limit? When is a metric space "complete" (i.e., not having any "holes" in it)? And, tying this topic to the other one, there are some interesting results about the cardinalities of some metric spaces.