These lecture notes grew out of notes I prepared for my lecture Topology held in
Erlangen in Summer Term 2012. The course is aimed at bachelor students in their
second year being familiar with the basic notions from calculus. The purpose of this
small course is to give some first introduction to the notions of general (or point set)
topology as they are needed in many other areas of mathematics. Of course, there
are many excellent textbooks on topology available. However, the aim of these
notes as well as of the lecture itself is to give bachelor students after their first year a
minimal amount of topology needed to continue with more advanced topics in a
mathematics (or physics) programme but still providing detailed proofs. Moreover,
the idea is to require only very basic preliminary knowledge as offered by the
introductory calculus and linear algebra courses.
The text is self-contained and provides many exercises which will enable the
student to work through these notes on her own. Alternatively, the text may serve as
a companion for a small lecture on topology.
The first four chapters can be seen as the core of the theory which every
mathematics student and hopefully also some physics students should be exposed
to. The remaining two chapters give a certain personal preference: I have chosen to
put some focus on possible applications in functional analysis. This explains why
the last two chapters are on continuous functions as well as on Baire’s Theorem in
different formulations. On the other hand, I have omitted other important concepts
like the fundamental groupoid or topological groups and their continuous actions
due to the lack of time and space.
The participants of the original lecture in Erlangen showed great patience with
the first versions, not only of these notes but also with the lecture itself. I would like
to thank all of them for their comments, remarks and suggestions, which all found
their way into these notes in one form or the other. In particular, I would like to
mention here Alexander Spies for numerous corrections and a careful proofreading
of the entire manuscript. Moreover, I am much obliged to Florian Unger for taking
care of the LATEX-files and all the typing of the first version of the draft. Without
his help, the manuscript would never have been finished. It is a pleasure to thank
Karl-Hermann Neeb for various discussions on the pedagogical aspects of teaching
of topology. The anonymous referees pointed out many weak points in the original
manuscript helping to improve it in many places. Their comments and remarks are
much appreciated. Last but not least, I would like to thank my family for the
patience and support throughout: without this the book would never have been
possible.