Tietze extension theorem


Differential geometry

differential geometry

synthetic differential geometry








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


For topological spaces


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}.


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.


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).


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).


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.


The smooth version is discussed in chapter I and II of

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