When studying abstract mathematics, one of the best ways to understand the subtleties that are inherent in the abstraction is to keep certain illustrative examples in mind. This allows us to encode the properties of the math we are studying in an easy to understand and familiar setting. When studying Rings, many think of the integers, or a ring of matrices. When studying continuous functions, many think back to the simple functions of introductory calculus. And when studying point set topology, many think of the familiar Euclidean spaces.
However, there is an inherent danger in making use of this technique in that one can often gloss over very important subtleties in the mathematics, which can be the difference between valid and incorrect mathematics in the end.
One excellent way to overcome this hurdle is to make use of counter examples.
The notion of a "counter example" is not precisely defined in mathematics, in part because it is a generic term which is applied to a number of different scenarios, and also likely because mathematicians don't see the need to formalize such language. However, a definition which encapsulates most counter examples would be:
Definition: A Counter Example is an example which illustrates how a given set of assumptions does not imply a certain conclusion.
The beauty of using counter examples is that they make us aware of the role of each of the assumptions in a given definition or theorem. They also provide us with valuable insight into the implications of a given definition or theorem, as well as what a definition or theorem does NOT say. Although these levels of insight and nuance can be obtained directly, it often occurs that the use of counter examples is a much more efficient way to illustrate and remember these subtleties. In the remainder of this article, we will examine several counter examples in my hope that the use of counter examples as a means of understanding will be clearly illustrated.
Continuity and Differentiability:
If you were to ask any reasonably versed calculus student what the relationship is between continuity and differentiability, most would be able to tell you that a differentiable function is always continuous. The more careful and discerning students could tell you, however, that a continuous function is not necessarily differentiable.
One might look to the function |x| to provide a counter example to illustrate that continuity at a point does not imply differentiability at a point, since |x| is not differentiable at the point x=0, even though it is continuous there.
However, there are in fact examples of functions which are continuous everywhere on their domain, but differentiable nowhere! These functions illustrate the fact that if a function is continuous, one only knows that your function MIGHT be differentiable (whereas if it were not continuous, one would immediately know the function was not differentiable).
Although the technique of Riemann Integration is powerful enough for many real world applications, it does still have significant drawbacks. One of the main drawbacks of the Riemann Integral is that it only works with function that are reasonably well behaved. "Reasonably well behaved", in this case, means that the points in the domain of f where f is discontinuities must not be too dense. Formally, this condition may be stated as:
Theorem: Let R be the set of real numbers, and let U be a measurable subset of R. A real valued function f:U->R, is Riemann Integrable if and only if the set of discontinuities of f has measure zero.
Thus, if we want to find a function which is not Riemann Integrable, a place to begin is by looking at very discontinuous functions.
Let f:[0,1]->R be defined by:
f(a)=1 if a is rational, and
f(a)=0 if a is irrational.
By inspection, we see this function is discontinuous on its entire domain. Taking the upper Riemann sums (Riemann sums using the supremum of the function on the given subintervals) yields
f(x_i)*x_i = 1 (where f(x_i)=sup f(x) on the interval x_i)
since sup f(x_i)=1 on all subintervals x_i (since the rationals are dense in [0,1]).
Similarly, the lower Riemann sums are given by:
f(x_j)*x_j = 0.
A function is Riemann integrable if and only if the upper Riemann sums equal the lower Riemann sums, demonstrating that this function is not Riemann integrable.
This demonstrates that the theory of Riemann integration is not suitable for dealing with many functions.
Compact Metric Spaces
Point set topology is a subject rich with examples which run counter to our intuition. The topology of R^n (Euclidean space) has a wealth of structures which makes the topology particularly nice. One property which makes Euclidean space nice is the fact that a set is compact if and only if it is closed and bounded.
However, this property is much more rare in general topological spaces, and in fact, even many metric spaces do not have the property that a set is compact if and only if it is closed and bounded.
To see this, consider the set Q intersect [0,1], where Q is the set of rational numbers. This set is a closed and bounded metric space. (Note that it is closed because it is an intersection of a closed set in R with Q.) However, it is not compact.
The easiest way to see this is to note that a metric space is compact if and only if every Cauchy sequence converges to a point in the space. (Note that if a Cauchy sequence does not converge to a point in the space, it is not said to converge.) Using this definition, we see that our space, Q intersect [0,1] is not compact (since there are Cauchy sequences which do not converge to a point in the space), even though the space is closed and bounded.
The above counter examples touch on three different topics which would be familiar to students with advanced undergraduate coursework. But this technique of utilizing counter examples to discern the subtle structure of mathematical objects is used throughout all of mathematics. For any definition and any theorem, there are counter examples you will find that help to illustrate what the definitions and theorems actually mean. They are an extremely important aspect of the study of higher math, and will help you in acquiring a thorough and deep understanding of even highly abstract mathematics.