Department of Mathematics

Syllabus

**Math 01.354 - Introduction to Topology**

**CATALOG DESCRIPTION:**

Math 01.354 Introduction to Topology
3 s.h.

(Prerequisites: Math 01.330 Introduction to Real Analysis I with a grade of C- or better)

This course includes the properties of general topological spaces, metric spaces, separation, compactness, connectedness and the Heine-Borel and Bolzano-Weierstrass theorems.

**OBJECTIVES:**

This one-semester three-credit course in Introductory Topology will have three general interconnected objectives. First, as it has become increasingly apparent that topology is one of the major branches of modern mathematics, this course will provide a firm foundation in topology to enable the student to continue more advanced study in this area. Second, as several important areas of mathematics, in particular modern analysis, depend upon or are clarified by the certain topics in topology, this course will present and emphasize those topics in order to aid the student in his future mathematical studies. Finally, this course hopes to expose the students to both mathematical rigor and abstraction, giving there an opportunity further to develop his mathematical maturity.

**CONTENT:**

1. **Sets and Relations**

1.1 Sets

1.2 Relations

1.3 Equivalence Relations

1.4 Composition of Relations

2. **Functions**

2.1 Functions

2.2 Indexed Sets

2.3 Cartesian Products

2.4 Associated Set Functions

2.5 Algebra of Real-Value Functions

3. **Topology of the Line and Plane**

3.1 Open Sets

3.2 Accumulation Points

3.3 Bolzano-Weierstrass Theorem

3.4 Closed Sets

3.5 Heine-Borel Theorem

3.6 Sequences

3.7 Completeness

4. ** Topological Spaces**

3.1 Open Sets and Topologies

3.2 Accumulation Points

2.3 Closure and Neighborhood

2.4 Bases and Subbases for a Topology

2.5 Coarser and Finer Topologies

2.6 Subspaces and Relative Topologies

2.7 Continuity

3. **Metric and Normed Spaces**

4.1 Distances and Metrics

4.2 Diameters and Open Spheres

4.3 Equivalent Metrics

4.4 Euclidean m-space

4.5 Hilbert Spaces

4.6 Normed Spaces

4. **Separation Axioms**

4.1 T1-Spaces

4.2 Hausdorff Spaces

4.3 Regular Spaces

4.4 Normal Spaces

5. ** Compactness**

5.1 Covers and Compact Sets

5.2 Finite Intersection Property and Compactness

5.3 Sequential Compact Sets

5.4 Compactness in Metric Spaces

6. **Connectedness**

6.1 Connectedness on R

6.2 Applications

6.3 Separated sets

**TEXTS:**

Buskes, Gerard :& Arnold VanRooij, TOPOLOGICAL SPACES FROM DISTANCE TO NEIGHBORHOOD, Springer Pub Co.

1. Sieradsky, Allan, AN INTRODUCTION TO TOPOLOGY AND HOMOTOPY, PWS; Kent Pub. Co. Boston, MA 1992.

2. Cain, George, INTRODUCTION TO GENERAL TOPOLOGY, Addison - Wesley,
Reading, MA, 1994.