Given a set of points X, a topology on X is some "extra-data" on X which allows us to make sense of statement such as: - two points of X are "close to each other"
- a sequence of points (in X) converges to another point.
- a function f: X --> |R is continuous
or other "topological statements" which you might have seen in analysis courses (such as compactness, connectedness). Familiar sets of points such as the circle, the sphere, the torus, although they are all "isomorphic" as sets (i.e. there are bijections between them), our intuition tells us that they look quite different. That is because our intuition sees not only the bare sets of points, but also the natural topologies on them (the meaning of "a point gets closer to another one" being clear in each of these examples). And our intuition is right: the circle, the sphere and the torus are not isomorphic as topological spaces.
Apart from "geometric examples" of topological spaces (circle, torii, cube, etc). you have probably seen (or you will see) other interesting examples coming from analysis, functional analysis, distribution theory, etc. Such as: the space of bounded continuous functions, the space of compactly supported smooth functions, the space of distributions on an open set in the Euclidean space.
Question A: A central question in topology is: given two topological spaces X and Y, how do we decide if they are isomorphic as topological spaces? Example: how do you PROVE that the circle and the sphere are not? Or that [0, 1] and [0, 1) are not? Question B: Another important question is: given two spaces X and Y, when can one obtain one from the other by "pushing" and "pulling" them, without "breaking them". Particular case: given a space X and a subspace A, when can one "push X inside A", without "breaking it", and without leaving X. Question B is often very useful for answering Question A, and it is at the origin of "Algebraic Topology".

The aim of the course is to make the students know, understand, and feel what topological spaces are, give them the tools to answer questions as the two posed above and, in particular, discuss in detail the case of surfaces.

The content of the course is as follows. The first few courses are dedicated to the notion of topology and immediate topological notions, to examples and to constructions which allow us to produce new examples out of old ones (direct product, direct sum, quotient topology, etc). Then we concentrate on deeper topological properties such as compactness, local compactness, connectedness, normality, which allow us to attack Question A in many examples. In the second part we discuss more refined tools for answering the two questions