Quiz 2 will cover the following topics.
- The Natural Numbers
- Mathematical Induction
- Well-Ordering Principle
- Ordered Fields
- Ordered Field Axioms (basic algebra with inequalities)
- Triangle Inequality
- All ordered fields contain \(\mathbb{Q}\)
- Examples: \(\mathbb{Q}\), \(\mathbb{R}\), hyperreal numbers
- The Completeness Axiom
- Upper and Lower Bounds
- Bounded Sets
- The Completeness Axiom (aka Least Upper Bound Principle)
- Supremums and Infimums
- \(\mathbb{Q}\) is Dense in \(\mathbb{R}\)
- The Archimedean Principle
- Topology
- Neighborhoods
- Interior Points and Boundary Points
- Open Sets and Closed Sets
- Accumulation Points and Isolated Points
- Closure of a Set
- Examples:
- All open intervals \((a,b)\) are open.
- All closed intervals \([a,b]\) are closed.
- All singleton sets \(\{x\}\) are closed.
- \(\mathbb{R}\) and \(\varnothing\) are both open and closed at the same time.
- Compact Sets
- Compact Sets
- Open Covers and Subcovers
- Heine-Borel Theorem
- Balzano-Weierstrass Theorem