 
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
