Alphabeta MathThe Tychonoff Embedding and the Stone–Čech 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
- 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
- Topological Spaces and Continuity
- Urysohn's Lemma and the Tietze Extension Theorem
2 · Logical flowchart
3 · Items
Definition
A family of continuous unit-interval-valued functions that separates points from closed sets
Definition
The evaluation map from a space into the unit cube indexed by a family of continuous functions
Lemma
The evaluation map of a point–closed-set separating family is a topological embedding
Theorem
A space is Tychonoff if and only if it embeds in a cube [0,1]^J
Definition
A Hausdorff compactification as a dense embedding into a compact Hausdorff space
Corollary
Assuming the ultrafilter lemma, every Tychonoff space has a Hausdorff compactification
Definition
The Stone–Čech compactification by its compact-Hausdorff extension property
Lemma
Every continuous [0,1]-valued function extends uniquely over the closure of the full evaluation image
Lemma
Under dependent choice, every compact Hausdorff space embeds in a unit cube
Theorem
Under the ultrafilter lemma and dependent choice, the closure of the full evaluation image is the Stone–Čech compactification
Corollary
Stone–Čech compactifications are uniquely homeomorphic over the original space
Corollary