Alphabeta MathComplete Metrizability, Čech-Completeness, and Baire Category: Examples and Counterexamples
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1 · Prerequisites
Abstract Algebra
Linear Algebra
Real Analysis
- Approximation and Compactness in C(K)
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- Topology of ℝ
Set Theory
Topology
- Compactness
- Compactness in Metric Spaces
- Complete Metrizability, Čech-Completeness, and Baire Category
- Completeness, Completion, and Uniform Continuity
- Convergence: Nets and Filters
- Countability Axioms and Cardinal Functions
- Hausdorff via the Diagonal
- Hereditary and Productive Behaviour of the Separation Axioms
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Tychonoff Embedding and the Stone–Čech Compactification
- Topological Spaces and Continuity
- Urysohn's Lemma and the Tietze Extension Theorem
2 · Logical flowchart
3 · Items
False statement
FALSE: every Baire space is completely metrizable
False statement
FALSE: the rational numbers form a Baire space
Example
Cylinder sets and continued fractions exhibit the homeomorphism ℕ^ℕ≅ℝ∖ℚ
Example
Under the Axiom of Choice, the Hilbert cube is compact, Polish, and universal for separable metrizable spaces
Corollary
Under Dependent Choice, nowhere differentiable functions form a residual subset of C([0,1],ℝ)
False statement