Topology

COURSE OBJECTIVE
At the end of this course:
-the student is familiar with the basic concepts, such as topology, openand closed sets, and continuous functions, and is able to use these toprove some fundamental results;
-the student knows several ways to define a topology on a set and cancompare different topologies on the same set;
-the student can determine if a given topological space satisfiescertain properties (such as compactness and connectedness);
-the student understands the concept of topological manifold andfundamental group, and is aware of the importance of these concepts for
more advanced subjects, such as algebraic topology and differential geometry.

COURSE CONTENT
This course is a first introduction to topology, that is, the
mathematical study of the shape of space. The concepts that are
introduced in this course are essential in understanding more advanced
subjects, such as differential topology, functional analysis, and
algebraic topology.

The following topics will be covered during the course:
-general topological spaces;
-topology generated by a basis;
-continuous maps and homeomorphisms;
-connectedness, path-connectedness, local connectedness;
-compactness, local compactness;
-products and quotients;
-topological manifolds;
-separation axioms;
-the fundamental group of a topological space.

TEACHING METHODS
Lectures and tutorials (2+2 hours per week)

TYPE OF ASSESSMENT
For this course there are 4 hand-in assignments (total: 20%), a midtermexamination (30%) and a final exam (50%). There will also be a resitexamination: the final grade will then be determined by the grade of the resit (80%) and the grade of the hand-in assignments (20%).

RECOMMENDED BACKGROUND KNOWLEDGE
Basic concepts of mathematics, Analysis 1 and 2, Group theory