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:
- Munkres, J.R. (2015). Topology, a first course, Prentice-Hall, India.
- Janich, K. (1984). Topology, Springer.
- Armstrong, M.A. (1983). Basic Topology, Springer.
- Morris, S. A. Topology Without Tears. https://www.topologywithouttears.net/topbook.pdf