Alphabeta MathMetrization: Urysohn, Nagata–Smirnov, Bing, Smirnov: 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
- 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
- Countability Axioms and Cardinal Functions
- Discrete bases in metric spaces
- Hereditary and Productive Behaviour of the Separation Axioms
- Metric Spaces
- Metrization: Urysohn, Nagata–Smirnov, Bing, Smirnov
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- Topological Spaces and Continuity
2 · Logical flowchart
3 · Items
Example
An uncountable discrete space is metrizable and has a discrete basis, but is not second countable
Example
Under choice, the lower-limit line is regular and separable but not second countable and therefore not metrizable
False statement
FALSE: every regular space is metrizable
Example