# nLab Tietze extension theorem

### Context

#### Differential geometry

differential geometry

synthetic differential geometry

# Contents

## 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 $f : A \to \mathbb{R}$ to the real line a continuous function $F : X \to \mathbb{R}$ extending it, i.e. such that $F|_A = f$.

### For smooth loci

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

###### Definition

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

###### Lemma

Let $X$ be a smooth manifold and let $\{g_i \in C^\infty(X)\}_{i = 1}^n$ 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 $(d g_i) : T_x X \to \mathbb{R}^n$ 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 Chapter I of (MoerdijkReyes).

###### Proposition

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

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

###### Proof

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

## 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)