Math 553: Foundations of Topology 1

Catalog Information

Title

Foundations of Topology 1.

(Credit Hours:Lecture Hours:Lab Hours)

(3:3:0)

Offered

F

Description

Naive set theory, topological spaces, product spaces, subspaces, continuous functions, connectedness, compactness, countability, separation axioms, metrization, and complete metric spaces.

Desired Learning Outcomes

Students should gain a familiarity with the general topology that is used throughout mathematics.

Minimal learning outcomes

1. Set Theory
   • Finite, countable, and uncountable sets
   • Well-ordered sets
2. Topological Spaces
   • Basis for a topology
   • Product topology
   • Metric topology
3. Continuous Functions
4. Connectedness
5. Compactness
   • Tychonoff Theorem
6. Countability and Separation Axioms
   • Countable basis
   • Countable dense subsets
   • Normal spaces
   • Urysohn Lemma
   • Tietze Extension Theorem
7. Metrization
   • Urysohn Metrization Theorem
8. Complete Metric Spaces

Textbooks

Possible textbooks for this course include (but are not limited to):

Additional topics

Paracompactness, the Nagata-Smirnov Metrization Theorem, Ascoli's Theorem, Baire Spaces and dimension theory as time allows.

Courses for which this course is prerequisite

Math 554