Math 446646 Review for Test 1 Summary: Math 446­646 Review for Test 1 Concepts you must master for this test (including their proofs) Chapter 2: Metric Spaces · Definition of metric space and properties of a metric or distance. Examples of metric spaces seen in class and in the exercises of section 2. Theorem 2.3: How to define a metric on the product of metric spaces. · All the equivalent ways of proving that a function between metric spaces is continuous: Def 3.2, Theorems 4.2, 4.3, 4.6, 4.7, 5.4, 6.3, 6.10. Examples of continuous functions: Theorems 3.3, 3.4, 3.5, Exercises 3.1, 3.2, 3.3. · Open balls and neighborhoods: Def 4.1, 4.4, Theorems 4.5, 4.8. · Limits: Def 5.2, Corollary 5.3, Lemma 5.6, Corollary 5.7. · Open sets: definition 6.1 and properties (Th. 6.2, 6.3). Closed sets and limit points: Def 6.5, 6.6, and properties (Th. 6.7, 6.8, 6.11). · Metrically equivalent spaces and Topologically equivalent spaces: Defs 7.3, 7.6, Theorems 7.4, 7.5, 7.7, 7.8, 7.9), examples, exercise 2.3. Chapter 3: Topological Spaces · Section 2: Definition of topological space, examples seen in class, basis of a topology, how to compare two topologies if we know their bases. · Section 4: Open and closed sets, interior, closure and boundary of a Collections: Mathematics