Tietze Extension Theorem

The Tietze extension theorem states that continuous real-valued functions defined on closed subsets of a normal space can be extended to the entire space. This is one of the most powerful and widely applied results in topology and analysis.

Statement

Theorem6.9Tietze Extension Theorem

Let $X$ be a normal topological space and $A \subseteq X$ a closed subset. If $f: A \to [a, b]$ is continuous, then there exists a continuous extension $F: X \to [a, b]$ with $F|_A = f$ .

More generally, if $f: A \to \mathbb{R}$ is continuous, then there exists a continuous extension $F: X \to \mathbb{R}$ with $F|_A = f$ .

Proof Outline

Proof

We prove the bounded case $f: A \to [-1, 1]$ ; the general case follows by rescaling and composing with a homeomorphism $\mathbb{R} \cong (-1, 1)$ .

Step 1: Approximate $f$ by a function on $X$ .

Define $B_- = f^{-1}([-1, -1/3])$ and $B_+ = f^{-1}([1/3, 1])$ . These are disjoint closed subsets of $A$ , hence of $X$ (since $A$ is closed). By the Urysohn lemma, there exists $g_1: X \to [-1/3, 1/3]$ with $g_1|_{B_-} = -1/3$ and $g_1|_{B_+} = 1/3$ .

Then $|f(x) - g_1(x)| \leq 2/3$ for all $x \in A$ .

Step 2: Iterate.

Apply Step 1 to $f - g_1|_A: A \to [-2/3, 2/3]$ (after rescaling) to get $g_2: X \to [-2/9, 2/9]$ with $|f - g_1|_A - g_2|_A| \leq 4/9 = (2/3)^2$ .

Continuing, we get a sequence $g_n: X \to [-(2/3)^{n-1}/3, (2/3)^{n-1}/3]$ such that: $\left|f(x) - \sum_{k=1}^n g_k(x)\right| \leq \left(\frac{2}{3}\right)^n \quad \text{for all } x \in A.$

Step 3: Define the extension.

Set $F(x) = \sum_{n=1}^{\infty} g_n(x)$ . The series converges uniformly (by comparison with $\sum (2/3)^{n-1}/3 = 1$ ). The uniform limit of continuous functions is continuous.

For $x \in A$ : $F(x) = \lim_{N \to \infty} \sum_{k=1}^N g_k(x) = f(x)$ (since the partial sums converge to $f$ on $A$ ).

The bound $|F(x)| \leq \sum_{n=1}^{\infty} (2/3)^{n-1}/3 = 1$ ensures $F: X \to [-1, 1]$ .

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Consequences

Theorem6.10Equivalent Characterization of Normality

For a $T_1$ space $X$ , the following are equivalent:

$X$ is normal.
(Urysohn lemma) Disjoint closed sets can be separated by continuous functions.
(Tietze) Every continuous $f: A \to \mathbb{R}$ from a closed $A \subseteq X$ extends to $F: X \to \mathbb{R}$ .

Proof

(1 $\Rightarrow$ 2) is the Urysohn lemma. (2 $\Rightarrow$ 1) is noted above. (1 $\Rightarrow$ 3) is the Tietze theorem. (3 $\Rightarrow$ 2): Given disjoint closed $A, B$ , define $f: A \cup B \to [0, 1]$ by $f|_A = 0$ and $f|_B = 1$ . This is continuous (on a closed subset). Extend to $F: X \to [0, 1]$ .

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Applications

ExampleExtending Functions from Submanifolds

If $M$ is a closed submanifold of a normal space $X$ (e.g., a compact manifold inside $\mathbb{R}^n$ ), any continuous function on $M$ extends to $X$ . This is used in differential topology to extend smooth functions via approximation.

ExampleThe Tietze Theorem in Metric Spaces

For a metric space $(X, d)$ with closed $A \subseteq X$ and continuous $f: A \to \mathbb{R}$ , the McShane--Whitney extension provides an explicit formula: $F(x) = \inf_{a \in A} \{f(a) + L \cdot d(x, a)\}$ where $L$ is the Lipschitz constant of $f$ (if $f$ is Lipschitz). For general continuous $f$ , the Tietze construction via the Urysohn lemma is needed.

ExampleApplications in Functional Analysis

The Tietze theorem is used to prove:

Partitions of unity (subordinate to open covers of normal spaces).
Approximation results: Stone--Weierstrass type theorems often require extending functions.
Dugundji extension theorem: A generalization for metric spaces where the extension preserves convex combinations.

RemarkLimitations

The Tietze theorem requires $A$ to be closed. For open subsets, extension is generally impossible without additional conditions. For instance, $f: (0, 1) \to \mathbb{R}$ given by $f(x) = \sin(1/x)$ has no continuous extension to $[0, 1]$ .

Also, the theorem applies to real-valued (or $\mathbb{R}^n$ -valued) functions. Extension to spaces other than $\mathbb{R}^n$ requires the target to have special properties (absolute neighborhood retract, or ANR).