Topology

Course Code :1003WETATP
Study domain:Mathematics
Academic year:2017-2018
Semester:2nd semester
Sequentiality:
Contact hours:30
Credits:3
Study load (hours):84
Contract restrictions: No contract restriction
Language of instruction:Dutch
Exam period:exam in the 2nd semester
Lecturer(s)Werner Peeters

3. Course contents *

The first part of the course consists of:

  • an enumeration of the five equivalent descriptions of a topology (with open and closed sets, neighbourhoods, the interior and closure operator),
  • a study of the notions of convergence and adherence in a general topological space by means of filters and ultrafilters,
  • the equivalent descriptions of continuity in a topological space,
  • initial and final structures, with peculiar attention to products and subspaces on the one hand, and quotient spaces and (to a lesser extent) disjoint sums on the other.

In the second part, an overview will be given of the topological properties, i.e. poperties that are preserved onder homomorphisms, and an extensive classification will be made. We will distinguish:

  • countability properties: A2, A1 and separability
  • separation axioms: T0 (Kolmogorov), T1 (Fréchet), T2 (Hausdorff), regularity, T3, complete regularity, T3 1/2, normality, T4 and the Urysohn lemma, as well as some important corollaries such as the closed graph theorem
  • compactness properties: compactness, sequential compactness, local compactness the Alexandroff-compactification, Lindelöf, sigma-compact, Baire spaces and countable compactness
  • Connectedness properties