Math 553: Foundations of Topology 1

Catalog Information

Title

Foundations of Topology 1.

(Credit Hours:Lecture Hours:Lab Hours)

(3:3:0)

Offered

F

Prerequisite

Math 341 or equivalent.

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