Instructor: Dr. Alex KÜRONYA
Text: the official text for the course is the set of lecture
notes written by László Fehér. It is available here,
or may be bought at the office.
Recommended books: Bredon: Topology and Geometry, Springer, 1997
Munkres: Topology, Prentice Hall, 2000 (2nd edition)
Hatcher: Algebraic topology, Cambridge University Press, 2002
Of these the first two are interchangable for our purposes. Both are very well written, with Munkres giving more details in general. The book of Hatcher is a good and very detailed introduction to algebraic topology, however, it covers only the last part of the course. At the moment it is still available online from the author's website.
Prerequisite: Calculus, especially metric spaces, the notion of continuity, basics of set theory. The definition and basic properties of groups will also be needed during the second part of the course, but this can also be learned quickly in the form of supplementary reading.
Course description: This is a standard introductory course on point-set topology and the rudiments of algebraic topology, roughly equivalent to a first year graduate course on the subject. Our purpose here is to get acquainted with basic concepts of the field. For the most part, the course will be devoted to general topology: the topics covered include metric and topological spaces, continuity, homeomorphisms, construction of topologies, connectedness, compactness, and separation axioms (among many others). Along the way we will necessarily study numerous applications and examples, mostly coming from geometry. The particular applications we consider will to some degree depend on the background of the class. In particular, if there is enough interest, one can go beyond the standard geometric circle of ideas and have a look at how topology arises in algebraic/arithmetic geometry.
In addition, we will make a quick excursion into algebraic topology.
The notion of the fundamental group of a topological space will be introduced,
and we will use it to study covering spaces and (if time permits) the classification
of compact surfaces.