Alphabeta MathCountability Axioms and Cardinal Functions: 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
- Continuity, IVT, EVT, and Uniform Continuity
- 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
- Series: Convergence and the Nonnegative Tests
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- Topology of ℝ
Set Theory
- Cardinal Arithmetic, Cofinality and the Alephs
- Construction of the Natural Numbers
- Order, Zorn's Lemma, and the Axiom of Choice
- Ordinal Arithmetic and the First Uncountable Ordinal
- Ordinals, Cardinals, and Transfinite Recursion
- Relations, Functions, and Quotients
- The ZFC Axioms and the Basic Set Constructions
Topology
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Countability Axioms and Cardinal Functions
- Hereditary and Productive Behaviour of the Separation Axioms
- Metric Spaces
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Topology of Euclidean Space
- Topological Spaces and Continuity
2 · Logical flowchart
3 · Items
Example
Under choice, for the usual real line, w=d=χ=L=c=ℵ₀ under the raw convention
Example
Under choice, for an infinite discrete space of cardinality κ, w=d=L=c=κ while χ=1
Example
For the lower-limit line, χ=d=L=c=ℵ₀ and w=2^ℵ₀ under choice
Example
Assuming choice, the lower-limit plane is first countable, separable, and ccc, but not second countable or Lindelöf
Example
The one-point compactification of the discrete real line is compact and Lindelöf but is neither first countable nor separable
Example
Assuming countable choice, ω₁ is first countable and countably compact but is not separable or Lindelöf
Example
Under choice, a concrete ccc nonseparable Cantor cube indexed above 2^ℵ₀
Counterexample
Assuming choice, a separable space with a nonseparable subspace: the lower-limit plane and its antidiagonal
Counterexample
Assuming choice, a Lindelöf space whose square is not Lindelöf: the lower-limit line
Sources
- D. H. Fremlin, Measure Theory, Chapter 5A
- Separable space (Wikipedia)
- Cardinal function (Wikipedia)
- UCR General Topology Notes
- Lower limit topology (Wikipedia)
- Sorgenfrey plane (Wikipedia)
- Fort space (Wikipedia)
- Alexandroff extension (Wikipedia)
- First uncountable ordinal (Wikipedia)
- Order topology (Wikipedia)
- Cantor cube (Wikipedia)