M367K TOPOLOGY I
Prerequisite and degree relevance: One of M361K or 365C with a grade of C- or better, or consent of instructor.
Course description: An introduction to topology, including sets, functions, cardinal numbers, and the topology of metric spaces
This is a course that emphasizes understanding and creating proofs. The number of topics required for coverage has been kept modest so as to allow instructors adequate time to concentrate on developing the students theorem proving skills. The syllabus below is a typical syllabus. Other collections of topics in topology are equally appropriate. For example, some instructors prefer to restrict themselves to the topology of the real line or metric space topology.
- Cardinality: 1-1 correspondance, countability, and uncountability.
- Definitions of topological space: basis, sub-basis, metric space.
- Countability properties: dense sets, countable basis, local basis.
- Separation properties: Hausdorff, regular, normal.
- Covering properties: compact, countably compact, Lindelof.
- Continuity and homeomorphisms: properties preserved by continuous functions, Urysohns Lemma, Tietze Extension Theorem.
- Connectedness: definition, examples, invariance under continuous functions.
Notes containing definitions, theorem statements, and examples have been developed for this course and are available. The notes include some topics beyond those listed above.