Alphabeta MathUniform Completeness, Completion, and the Samuel Compactification: Examples and Counterexamples
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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
- Limits of Real Functions
- 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 Completeness, Completion, and the Samuel Compactification
- Uniform Spaces: the Three Definitions
- Urysohn's Lemma and the Tietze Extension Theorem
2 · Logical flowchart
3 · Items
Example
Under dependent choice and the ultrafilter lemma, the Samuel compactification of a nonempty compact Hausdorff space adds no points up to unique uniform isomorphism
Example
Under dependent choice and the ultrafilter lemma, the Samuel compactification of the open unit interval is the closed unit interval
Example
Under dependent choice and the ultrafilter lemma, the Samuel compactification of the discrete natural numbers is beta N
Counterexample
The Samuel compactification map need not be a uniform embedding for the original uniformity
Example