Urysohn's Lemma and the Tietze Extension Theorem

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1 · Prerequisites

Abstract Algebra

Binary Operations, Monoids, Groups and Subgroups

Combinatorics

Finite Counting, Factorials and Binomial Coefficients

Linear Algebra

Real Analysis

Set Theory

Topology

2 · Logical flowchart

3 · Items

Definition

The dyadic rationals of [0,1], their finite levels Dₙ, and their density in [0,1]

Lemma

If (Uᵣ)_r ∈ D are open with Uᵣ ⊆ Uₛ whenever r < s and U₁ = X, then x ↦ inf{ r ∈ D : x ∈ Uᵣ } is a continuous map X → [0,1], and no choice principle is used

Theorem

Urysohn's lemma, under the axiom of dependent choice: in a normal space two disjoint closed sets are separated by a continuous function into [0,1], and conversely such a space is normal

Corollary

Under dependent choice a normal T₁ space is completely regular, so T₄ ⟹ T_31/2, and together with the implications already proved this is the whole classical chain

Lemma

If for every ε > 0 some continuous g : X → ℝ satisfies | f(x) - g(x)| < ε for all x, then f is continuous; in particular a uniformly convergent series of continuous real functions has a continuous sum

Theorem

Under dependent choice a space is perfectly normal if and only if it is normal and every closed set is a zero set

Theorem

Tietze's extension theorem, under dependent choice: a continuous map from a closed subspace of a normal space into [a,b] extends continuously to the whole space, and this property characterises normality

Corollary

Under dependent choice, a continuous real-valued map on a closed subspace of a normal space extends to the whole space, and a map into an open interval extends into that same open interval

Theorem

Under dependent choice a locally compact Hausdorff space is completely regular, hence Tychonoff

Corollary

Under dependent choice a compact Hausdorff space is Tychonoff, and its disjoint closed sets are separated by continuous functions

Remark

Urysohn's Lemma and the Tietze Extension Theorem

3 · Items

The dyadic rationals of [0,1], their finite levels Dₙ, and their density in [0,1]

If (Uᵣ)_r ∈ D are open with Uᵣ ⊆ Uₛ whenever r < s and U₁ = X, then x ↦ inf{ r ∈ D : x ∈ Uᵣ } is a continuous map X → [0,1], and no choice principle is used

Urysohn's lemma, under the axiom of dependent choice: in a normal space two disjoint closed sets are separated by a continuous function into [0,1], and conversely such a space is normal

Under dependent choice a normal T₁ space is completely regular, so T₄ ⟹ T_31/2, and together with the implications already proved this is the whole classical chain

If for every ε > 0 some continuous g : X → ℝ satisfies | f(x) - g(x)| < ε for all x, then f is continuous; in particular a uniformly convergent series of continuous real functions has a continuous sum

Under dependent choice a space is perfectly normal if and only if it is normal and every closed set is a zero set

Tietze's extension theorem, under dependent choice: a continuous map from a closed subspace of a normal space into [a,b] extends continuously to the whole space, and this property characterises normality

Under dependent choice, a continuous real-valued map on a closed subspace of a normal space extends to the whole space, and a map into an open interval extends into that same open interval

Under dependent choice a locally compact Hausdorff space is completely regular, hence Tychonoff

Under dependent choice a compact Hausdorff space is Tychonoff, and its disjoint closed sets are separated by continuous functions

Which results on this page spend dependent choice, which spend countable choice, and which are theorems of ZF

FALSE: Every normal space is completely regular

FALSE: Every continuous real-valued function on a subspace of a normal space extends continuously to the whole space