Alphabeta MathUniform Completeness, Completion, and the Samuel Compactification
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1 · Prerequisites
Real Analysis
- 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
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- Topology of ℝ
Set Theory
Topology
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Convergence: Nets and Filters
- Hausdorff via the Diagonal
- Hereditary and Productive Behaviour of the Separation Axioms
- Metric Spaces
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Tychonoff Embedding and the Stone–Čech Compactification
- Topological Spaces and Continuity
- Uniform Spaces: the Three Definitions
- Urysohn's Lemma and the Tietze Extension Theorem
2 · Logical flowchart
3 · Items
Definition
The Samuel uniformity generated by bounded uniformly continuous functions
Lemma
Samuel function pseudometrics generate a uniformity coarser than the original one
Lemma
The Samuel uniformity is totally bounded
Lemma
Assuming dependent choice, the Samuel uniformity induces the original topology
Definition
The Samuel completion and, when compactifying, the Samuel compactification
Lemma
Total boundedness passes to a uniform space with a dense uniformly continuous image
Theorem
Under the ultrafilter lemma the Samuel completion is compact, and under dependent choice plus the ultrafilter lemma it compactifies every separated uniform space
Theorem
Under dependent choice and the ultrafilter lemma, uniformly continuous maps to compact Hausdorff spaces extend uniquely over the Samuel compactification
Corollary
Samuel compactifications are unique up to the unique isomorphism fixing the original space
Lemma
Assuming dependent choice, a totally bounded uniformity equals its Samuel uniformity
Corollary
Under dependent choice the Samuel completion of a separated totally bounded space is its uniform completion; under the ultrafilter lemma it is compact
Corollary