Alphabeta MathUrysohn's Lemma and the Tietze Extension Theorem: 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
- Equivalent Forms of Completeness
- 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
- The Formal Laurent Series Field ℝ((t⁻¹)): Cauchy Complete, Non-Archimedean, Not Complete
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Example
A Urysohn function for (-∞, 0] and [1, ∞) in ℝ, written down and checked against the definition
Example
The sets U₀, U₁, U_1/2, U_1/4, U_3/4 of the Urysohn construction computed for two disjoint closed subsets of ℝ
Example
In a metric space the function d(x,A)/(d(x,A) + d(x,B)) separates two disjoint closed sets outright, so the metric case spends no choice principle
Example
Every closed subset of ℝ is a zero set and a G_δ, as the perfect-normality criterion predicts
Example
A continuous function on [0,1] ⊆ ℝ extended to all of ℝ, both by Tietze and by hand
Counterexample
The reciprocal on (0,1] is continuous and extends to no continuous function on ℝ, so closedness of the subspace is not decoration in the ℝ-valued Tietze extension
Example
Sierpinski space is normal and not completely regular, so the T₁ hypothesis in the Urysohn corollary is not decoration
Counterexample