# Math 553: Foundations of Topology 1

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## Contents

## Catalog Information

### Title

Foundations of Topology 1.

### (Credit Hours:Lecture Hours:Lab Hours)

(3:3:0)

### Offered

F

### Prerequisite

Mth 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

- Set Theory
- Finite, countable, and uncountable sets
- Well-ordered sets

- Topological Spaces
- Basis for a topology
- Product topology
- Metric topology

- Continuous Functions
- Connectedness
- Compactness
- Tychonoff Theorem

- Countability and Separation Axioms
- Countable basis
- Countable dense subsets
- Normal spaces
- Urysohn Lemma
- Tietze Extension Theorem

- Metrization
- Urysohn Metrization Theorem

- 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