Topology
Topology is a significant part of mathematics that focuses on spatial properties preserved under continuous deformations of objects, like stretching. (Tearing or gluing would not be allowed.) The subject was engendered from concepts in geometry and set theory, such as space, dimension, and transformation.
If you think the study of topology is far removed from the "real" world in which we live, you are mistaken. Applications of topological studies pervade all areas of our lives. For example, the surface of a golf ball has interesting topology, and manufacturers of golf balls hire mathematicians to do research on the design of golf ball surfaces that will lead to cutting edge performance through improved flight characteristics.
Topics that students in topology will study include:
- Set Theory and Logic
- Topological Spaces and Continuous Functions
- Connectedness and Compactness
- Countability and Separation Axioms
- The Tychonoff Theorem
- Metrization Theorems and Paracompactness
- Complete Metric Spaces and Function Spaces
- Baire Spaces and Dimension Theory
- Algebraic Topology
- The Fundamental Group
- Separation Theorems in the Plane
- The Seifert-van Kampen Theorem
- Classification of Surfaces
- Classification of Covering Spaces
- Applications to Group Theory
MIT's Open Courseware has several entire courses in topology available to students for free, including Introduction to Topology. Many books on this subject can be found at Google Books and Amazon.com. Good tutorials on topology are available through the University of Illinois at Urbana-Champaign's Beckman Institute for Advanced Science and Technology, Topology Tutorial I, and SUNY Stony Brook's Topology Tutorial II. Journals that students should be following include Oxford University Press' Journal of Topology and Elsevier's Topology.
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