Let us start with some history.
Controversy erupted in mathematics in the 1800's over the foundations of mathematics. There had long been controversial issues:
The axiomatic approach to mathematics was instituted to try to settle these sorts of controversies definitively, to make it clear what mathematics was about , and to put mathematics on a firm foundation. The goal was to reduce the need for intuition in deciding the truth of mathematical statements. Intuition is still needed to find proofs, isolate interesting ideas, etc. But, we don't want to be arguing about what a function is when we're trying to prove that, for example, a continuous function on a closed, bounded interval has a maximum and minimum value.
This axiomatic approach turned out to be harder than expected. Russell's paradox showed that reasonable looking methods of defining sets easily lead to impossible difficulties. The solutions to this have led to an immensely greater understanding of logic itself, with some celebrated theorems such as Godel's Incompleteness Theorem, which says roughly that there are statements of arithmetic which cannot be proven to be true or false, so that our theories of arithmetic are inherently incomplete.
The axiomatic approach has also paid off in every field of mathematics. The immense power of the axiomatic theories which were developed in the 20-th Century has led to the solution of old problems like Fermat's last theorem. These abstract theories are now available as tools with which we can probe old problems much more deeply.
Much of the mathematics you have studied up to this point has consisted primarily of calculations. Now the focus will be on ideas. However, calculation has its virtues, chief among them its universality and unassailability. If I claim that the quintessential characteristic of humans is their seach for knowledge (viz., homo sapiens ), we can get into long and difficult arguments over the truth of this claim. But when I claim that 2+2=4, we are unlikely to disagree, especially once we have carefully defined 2, +, and 4. Nor will there be any disagreement with the claim that the continuous image of a connected set is connected.
Formal logic gives us a way of calculating with ideas and sorting through their consequences to find connections which are not evident on first inspection. The elementary part of formal logic, propositional calculus , is complete and decidable in a simple algorithmic way known as truth tables . Usually, we use simple rules of calculation such as
but it is comforting to know that we can always resort to truth tables to check something if we need to.
The more sophisticated forms of logic involve quantifiers. Suppose we are
given a set A and a statement P(x) about the elements x in A. Then we have
the existential quantifier
For example, we could say
which is false, or we could say
which is true. Or we could say
which is true, or we could say
which is false. There are calculation rules for these as well, and they will be very useful. For example,
and
Unfortunately, once we admit quantifiers into our language, we lose the ability to determine truth and falsity by simple calculations in general, so ingenuity is required. Further, Godel showed that (given certain simple axioms) there will be statements which we cannot prove either true or false, further complicating matters. One of these is the
Continuum Hypothesis: Any subset of the real numbers can be put into a one to one correspondence with either a subset of the integers or the whole real numbers.
In other words, te Continuum Hypothesis says that there are no intermediate sizes in the sense of cardinality, larger than the integers but smaller than the reals.
Fortunately, most of what we care about does not depend upon these undecidable statements.
Geometry was a precursor of topology. However, geometry focuses on issues of size and distance, whereas topology does not. For example, if we have a continuous function defined on the interval [1,2] whose value at 1 is 0 and whose value at 2 is 10, then we can be certain that somewhere between 1 and 2 the function will take on the value 5.
Similarly, if we have a continuous function defined on the interval [1000,2000] whose value at 1000 is 0 and whose value at 2000 is 10, then we can be certain that somewhere between 1000 and 2000 the function will take on the value 5. The fact that the interval [1000,2000] is 1000 times as large as the interval [1,2] has no bearing on this question.
For another example, if we have a continuous function defined on a sphere whose value at the North pole is 0 and whose value at the South pole is 10, then we can again be certain that somewhere on the sphere, there will be a point where the function takes on the value 5. All that matters in these three examples is that the domain of the function is connected.
Exercise: Define a function on [1,2] whose value at 1 is 0 and whose value at 2 is 10, but which never takes on the value 5.
Another example of a situation where topology matters more than geometry is a subway map. It is less important to show the actual distances between the stations than it is to show where one can switch from one subway line to another. That is, the topological issue, the way in which the subway lines are connected, is more important than the geometric issue of distance.
Here is another example. A continuous function defined on [1,2], or on [1000,2000], or on the sphere, must take on a maximum value and a minimum value. However, a function defined on the open interval (1,2) or (1000,2000), or on the sphere with the North pole removed, need not. The distinction here is that the forme three spaces are compact, while the latter three are not.
Exercise: Define continuous real valued functions on the latter three spaces which fail to have a maximum value.
Two of the most fundamental ideas in topology are connectedness and compactness . We shall focus on these and their relatives in this introductory course.
Finally, I want to point out some relations between topology and other branches of mathematics. All of these fields have contributed to the development of the field of topology, building more complicated structures on top of underlying topological spaces, and using the topological properties of those spaces to help analyze these additional structures.
Differential topology studies manifolds, spaces on which it is possible to say not only that something is continuous, but that it is smooth (i.e., differentiable). Many spaces which occur in other fields are manifolds, and these sorts of spaces have many special properties, such as Poincare duality.
Riemannian geometry studies a sophisticated form of geometry which can occur on a manifold, in which the notion of distance can vary in subtle ways from point to point. This proved to be the key concept Einstein needed to formulate the ideas of general relativity.
Functional analysis treats collections of functions as topological spaces, so that relations between functions and existence of particular types of functions can be studied in much the same way relations between numbers or vectors can be studied. For example, if I have a connected space of integrable functions containing a function whose integral is 0 and another whose integral is 10, then just as above, I can conclude there is a function whose integral is 5, at least if the process of integration is a continuous one.
Algebraic geometry is a way of treating algebraic equations as a form of topology. The topology in question here, the Zariski topology, very accurately reflects algebraic relations. It is a very odd topology at first encounter, in comparison with the topologies we are familiar with in the line, the plane, etc. It has proven to be quite a powerful tool, however, playing an important role in the proof of Fermat's last theorem, for example.
Algebraic topology goes the other way: converting topological questions into algebraic ones. This has the virtue of turning questions about spaces with very many points into questions of algebra, which are often more discrete and tractable. For example, the proof using algebraic topology of the Brouwer Fixed Point Theorem, that any continuous function from an n-dimensional ball to itself must have a fixed point, reduces to the simple algebraic fact that the identity function from the integers to the integers is not a constant function.
Of course, none of these fields operates in isolation. There are strong interactions between them in many different directions which you will learn as you progress in your studies.