nLab
Tietze extension theorem

Summary

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

Statement

For topological spaces

Theorem

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

For smooth manfiolds

See Steenrod-Wockel approximation theorem.

For smooth loci

Let $𝕃=(C^{\mathrm{\infty }}{\mathrm{Ring}}^{\mathrm{fin}}{)}^{\mathrm{op}}$ 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 $\hookrightarrow 𝕃$.

Definition

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

Lemma

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

This is lemma 2.1 in (MoerdijkReyes).

Proposition

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

This is prop. 1.6 in (MoerdijkReyes).

Proof

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

Related theorems

• the Hadamard lemma

• Borel's theorem

• the Steenrod-Wockel approximation theorem

References

The smooth version is discussed in chapter I and II of

• Ieke Moerdijk, Gonzalo Reyes, Models for Smooth Infinitesimal Analysis