Alphabeta MathHereditary and Productive Behaviour of the Separation Axioms: 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
2 · Logical flowchart
3 · Items
Example
A worked preservation table for subspaces and products of the standard, cofinite, and ordinal topologies
Example
Closed subspaces inherit normality: compact intervals, ordinal endpoints, and finite closed pieces
Example
Assuming countable choice, the deleted Tychonoff plank worked as T₃ but not normal inside its compact Hausdorff normal parent
Example
Assuming choice, the lower-limit line is normal while its square is regular and nonnormal
Counterexample
Assuming countable choice, normality is not open-hereditary: deleting one corner of the Tychonoff plank
Counterexample
Assuming choice, normality is not even finitely productive: two copies of the lower-limit line
Counterexample
Regular and normal do not imply T₁ under the library's conventions
Sources
- J. P. May, An Outline Summary of Basic Point Set Topology, §6
- Separation axiom (Wikipedia)
- J. R. Munkres, Topology, 2nd ed., §31
- Normal space (Wikipedia)
- Tychonoff plank (Wikipedia)
- L. A. Steen and J. A. Seebach, Counterexamples in Topology, deleted Tychonoff plank
- G. Gruenhage, General Topology Course Notes, Sorgenfrey plane
- Sorgenfrey topology (Encyclopedia of Mathematics)
- Sorgenfrey plane (Wikipedia)
- S. Willard, General Topology, separation axioms
- Sierpinski space (Wikipedia)