nLab
Tietze extension theorem

Context

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Contents

Summary

The Tietze extension theorem says that continuous functions extend from closed subsets of a normal space XX to the whole space XX.

Statement

For topological spaces

Theorem

For XX a normal topological space and AXA \subset X a closed subset, there is for every continuous function f:Af : A \to \mathbb{R} to the real line a continuous function F:XF : X \to \mathbb{R} extending it, i.e. such that F A=fF|_A = f.

For smooth manifolds

See Whitney extension theorem, also Steenrod-Wockel approximation theorem.

For smooth loci

Let 𝕃=(C Ring fin) op\mathbb{L} = (C^\infty Ring^{fin})^{op} be the category of smooth loci, the opposite category of finitely generated generalized smooth algebras. By the theorem discussed there, there is a full and faithful functor Diff 𝕃\hookrightarrow \mathbb{L}.

Definition

For A=C ( n)/JA = C^\infty(\mathbb{R}^n)/J and B=C ( n)/IB = C^\infty(\mathbb{R}^n)/I with IJI \subset J and BAB \to A the projection of generalized smooth algebras the corresponding monomorphism AB\ell A \to \ell B in 𝕃\mathbb{L} exhibits A\ell A as a closed smooth sublocus of B\ell B.

Lemma

Let XX be a smooth manifold and let {g iC (X)} i=1 n\{g_i \in C^\infty(X)\}_{i = 1}^n be smooth functions that are independent in the sense that at each common zero point xXx\in X, i:g i(x)=0\forall i : g_i(x)= 0 we have the derivative (dg i):T xX n(d g_i) : T_x X \to \mathbb{R}^n is a surjection, then the ideal (g 1,,g n)(g_1, \cdots, g_n) coincides with the ideal of functions that vanish on the zero-set of the g ig_i.

This is lemma 2.1 in Chapter I of (MoerdijkReyes).

Proposition

If AB\ell A \hookrightarrow \ell B is a closed sublocus of B\ell B then every morphism AR\ell A \to R extends to a morphism BR\ell B \to R

This is prop. 1.6 in Chapter II of (MoerdijkReyes).

Proof

Since we have R=C ()R = \ell C^\infty(\mathbb{R}) and C ()C^\infty(\mathbb{R}) is the free generalized smooth algebra on a single generator, a morphism AR\ell A \to R is precisely an element of C ( n)/JC^\infty(\mathbb{R}^n)/J. This is represented by an element in C ( n)C^\infty(\mathbb{R}^n) which in particular defines an element in C ( n)/IC^\infty(\mathbb{R}^n)/I.

References

The smooth version is discussed in chapter I and II of

Revised on January 16, 2014 23:15:00 by Anonymous Coward (93.40.105.214)