Department of Mathematics

Syllabus

**Math 01.526 - Point Set Topology**

**CATALOG DESCRIPTION:**

**Math 01.526 Point Set Topology
3 s.h.**

An introduction to one of the major branches of modern mathematics covering
axiomatic development of topological spaces and metric spaces, and the
concepts of convergence, continuity, separation, compactness and connectedness.

**OBJECTIVES:**

It is the purpose of this course to introduce the student to one of
the major branches of modern mathematics. As such, it is of itself worthy
of his attention. In addition, several important areas of mathematics,
in particular modern analysis, including elementary calculus, depend upon
or are clarified by certain topological concepts. To this end the course
is designed to emphasize these concepts and to advance his mathematical
maturity by exposing the student to other mathematical rigor and abstraction.

**CONTENT:**

**1. Introduction**

1.1 Set theory

1.2 Indexing notation

1.3 Function theory

**2. Topologies and Topological Spaces**

2.1 Open and closed sets

2.2 Metric spaces

2.3 Neighborhoods

2.4 Bases for a topology

2.5 Closure, interior and boundary

2.6 Continuity

2.7 Homeomorphism

2.8 Separation axioms - T sub 0, T sub 1, T sub 2

**3. Connectedness**

3.1 Connectedness or R

3.2 Applications

3.3 Separated sets

**4. Compactness**

4.1 Compact spaces

4.2 Compact subsets of R

4.3 Heine-Borel Theorem

**TEXTS:**

*Barnsley, Michael, *FRACTALS EVERYWHERE,* Academic Press, 2nd edition

Roseman, Dennis, *ELEMENTARY TOPOLOGY, *Prentice Hall, Upper Saddle River,
NJ 1999.

Buskes, G and Rooij, Aivan, *TOPOLOGICAL SPACES, *Springes, NY, NY,*
*1997.