Description
Book SynopsisOffers a comprehensive introduction to point-set, algebraic, and geometric topology, designed to support inquiry-based learning (IBL) courses for upper-division undergraduate or beginning graduate students. The book presents an enormous amount of topology, allowing an instructor to choose which topics to treat.
Table of Contents
- Introduction: The enchanting world of topology
- Point-set topology: Cardinality: To infinity and beyond
- Topological spaces: Fundamentals
- Bases, subspaces, products: Creating new spaces
- Separation properties: Separating this from that
- Countable features of spaces: Size restrictions
- Compactness: The next best thing to being finite
- Continuity: When nearby points stay together
- Connectedness: When things don't fall into pieces
- Metric spaces: Getting some distance
- Algebraic and geometric topology: Transition from point-set topology to algebraic and geometric topology: Similar strategies, different domains
- Classification of 2-manifolds: Organizing surfaces
- Fundamental group: Capturing holes
- Covering spaces: Layering it on
- Manifolds, simpleces, complexes, and triangulability: Building blocks
- Simplicial $\mathbb{Z}_2$-homology: Physical algebra
- Applications of $\mathbb{Z}_2$-homology: A topological superhero
- Simplicial $\mathbb{Z}$-homology: Getting oriented
- Singular homology: Abstracting objects to maps
- The end: A beginning--reflections on topology and learning
- Appendix: Group theory background
- Index
