PMAT 41403: Topology

Course Code    : PMAT 41403

Title                   : Topology

Pre-requisites   : PMAT 21263

 Learning Outcomes:

By the end of this course, the student should be able to,

1. analyze and interpret basic topological concepts introduced in this course

2. discuss and work with abstract topological spaces and develop tools to characterize them at a depth suitable for someone aspiring to study higher-level mathematics

3. formulate tools to identify when two topological spaces are equivalent (homeomorphic)

4. differentiate between functions that define a metric on a set and those that do not

5. describe the hereditary of topological properties under continuous maps

6. summarize the basis of point-set topology.

 Course Contents:

Topological spaces: 

• Definition, Open sets, closed sets, Basis (any topology/given topology), Finite-Closed topology, Euclidean topology.

Limit Points: 

• Definition, Boundary, Closure, Dense sets, Neighborhoods, Connectedness.

Homeomorphisms: 

• Definition, Subspaces, Non-Homeomorphic spaces, Hausdorff space.

Continuous Mappings: 

• Definition, Intermediate Value Theorem.

Metric Spaces: 

• Definition, Convergence (of a sequence), Completeness, Contraction Mappings, Baire Spaces.

Compactness: 

• Definition, Heine-Borel Theorem.

Product Topology

 Method of Teaching and Learning : A combination of lectures, problem set discussions and presentations.

 Assessment :  Based on problem sets, presentations and end of course examination  

Recommended Reading:

1. Munkres, J.R. (2015). Topology, a first course, Prentice-Hall, India.

2. Janich, K. (1984). Topology, Springer.

3. Armstrong, M.A. (1983). Basic Topology, Springer.

4. Morris, S. A. Topology Without Tears. https://www.topologywithouttears.net/topbook.pdf