Alphabeta MathPartitions of Unity and Paracompactness: 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
- Connectedness
- Hereditary and Productive Behaviour of the Separation Axioms
- Metric Spaces
- Partitions of Unity and Paracompactness
- 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
Example
A locally finite hat-function partition of unity on ℝ subordinate to overlapping intervals
Example
Under choice and dependent choice, a finite subordinate partition of unity for a two-set cover of a compact interval
Example
Point-finite need not mean locally finite: intervals accumulating at the origin
Counterexample
Without local finiteness, a pointwise finite sum of continuous functions can be discontinuous
Example
Assuming choice, ω₁ is countably compact, noncompact, and not paracompact
Counterexample
Assuming choice, paracompactness is not open-hereditary: ω₁ inside ω₁+1
Counterexample
Assuming choice, two paracompact lower-limit lines can have a nonparacompact product
Sources
- J. Robbin, Partitions of Unity
- K. Datchev, Iterated interpolation and a partition of unity (Purdue University)
- Locally finite collection (Wikipedia)
- First uncountable ordinal (Wikipedia)
- G. Gruenhage, General Topology Course Notes
- M. Aitken, Compactness notes (California State University San Marcos)
- G. Gruenhage, General Topology Course Notes, Sorgenfrey plane and Jones's lemma
- R. Gardner, Notes on Munkres Section 41: Paracompactness (East Tennessee State University)
- Sorgenfrey topology (Encyclopedia of Mathematics)