The notion of continuity of functions of one or several real variables is a powerful one, that leads to derivatives, vector fields, differential equations and dynamical systems. The purpose of Topology is to define and study what is meant by "continuity" when the domain of a function is an arbitrary set, rather than some Euclidean space.
A topology on a set X is a specification of which subsets of X are open. The set X along with its topology is called a topological space.
In the first half of the course we will consider several well-known sets, and study the various topologies that they admit. In so doing, we will be led to generalize such notions as open ball, continuous map, connectedness, and compactness.
In the second half of the course, we will see how to translate topological problems into algebraic ones. For this, we introduce the idea of homotopy (or deformation) of continuous maps, and the fundamental group of a topological space.
We finish the course, hopefully, with a classification of surfaces, spaces that locally look like euclidean 2-space. Examples include euclidean 2-space itself, spheres, tori, and the projective plane.
There will be a five-minute quiz at the beginning of each lecture, starting January 11th. You will be asked to provide a definition or the statement of a theorem from the previous lecture. The goal is to encourage you to read your notes before each class. I am hoping that punctuality will be an added result. Knowing the vagaries of the Ottawa transit system, only fifteen quizzes (out of about 20) will count.
While there will only be four assignment, each one will be rather involved and will require a fair amount of reflection and work. The exercises will consist almost exclusively of proofs, and you will be evaluated on your logic and coherence. That is, if you write something confusing that somehow contains "the right idea", don't expect full points.
Assignments may be written in either English or French. I will be providing a translation dictionary of topological terms. Most of it is obvious.
Three hours. Definitions, statements of theorems, easy proofs, one proof from the notes.