# Difference between revisions of "Math 553: Foundations of Topology 1"

### Title

Foundations of Topology 1.

(3:3:0)

F

### Prerequisite

Math 451 or instructor's consent.

### 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):