M367K SyllabusTOPOLOGY I
Prerequisite and degree relevance: Mathematics 361K or 365C or consent of instructor.
Course description: This will be a first course that emphasizes understanding and creating proofs; therefore, it provides a transition from the problem-solving approach of calculus to the entirely rigorous approach of advanced courses such as M365C or M373K. The number of topics required for coverage has been kept modest so as to allow instructors adequate time to Concentrates 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.