Urysohn Lemma and Tietze Extension Theorem

These two theorems are the crown jewels of point-set topology, transforming the abstract property of normality into powerful, tangible tools for constructing and extending continuous functions. Understanding them is essential for progressing into algebraic topology, functional analysis, and geometry, as they provide the foundational glue that allows local data to be pieced together into well-behaved global objects. They answer a fundamental question: given a topological space, how much control do we have over its continuous real-valued functions?

Normal Spaces: The Required Stage

Before constructing functions, we must understand the stage on which these theorems perform. A topological space is called a normal space if it satisfies two conditions: first, it is $T_1$ (meaning every singleton set is closed), and second, any two disjoint closed sets can be separated by disjoint open neighborhoods. Formally, for disjoint closed sets $A$ and $B$, there exist open sets $U$ and $V$ such that $A\subseteq U$, $B\subseteq V$, and $U\cap V=\varnothing$.

Normality is the key separation axiom for Urysohn's Lemma and the Tietze Extension Theorem. Many "nice" spaces are normal: all metric spaces, all compact Hausdorff spaces, and all regular, second-countable spaces. However, normality is not hereditary (a subspace of a normal space need not be normal) and can be surprisingly fragile, which is why these theorems are so potent—they leverage this specific, strong property of the entire space.

Urysohn's Lemma: Constructing a Continuous Separator

Urysohn's Lemma provides the first major payoff for normality. It states: if $X$ is a normal topological space and $A$ and $B$ are disjoint closed subsets of $X$, then there exists a continuous function $f:X\to [0,1]$ such that $f(A)=\{0\}$ and $f(B)=\{1\}$. Such a function is said to separate the closed sets.

The proof is constructive and ingenious. Instead of trying to define $f$ directly, one uses normality to build a nested family of open sets indexed by the rational numbers in $[0,1]$. You start by using normality to get an open set $U_{1/2}$ such that $A\subseteq U_{1/2}\subseteq \overline{U_{1/2}}\subseteq X\setminus B$. This process is repeated recursively for dyadic rationals. The function is then defined for any $x\in X$ as: $$f(x)=\inf \{r\in \mathbb{Q}\cap [0,1]:x\in U_r\}.$$ The construction ensures this infimum is well-defined, and the careful nesting of the open sets guarantees the continuity of $f$. The function acts like a "continuous measure of distance" from the set $A$, even in spaces where a metric is not defined.

The Tietze Extension Theorem: Extending Functions

While Urysohn's Lemma builds a new function from scratch, the Tietze Extension Theorem extends an existing one. It states: let $X$ be a normal topological space and $A\subseteq X$ be a closed subset. Then any continuous function $f:A\to \mathbb{R}$ can be extended to a continuous function $F:X\to \mathbb{R}$. Furthermore, if $f$ is bounded, the extension can be chosen to preserve the same bounds (e.g., if $f:A\to [-M,M]$, then $F:X\to [-M,M]$).

The proof typically uses a sequence of approximations constructed via Urysohn's Lemma, showcasing how Urysohn's Lemma is often a tool used to prove Tietze. One common approach is to first prove it for bounded functions $f:A\to [-1,1]$. The idea is to iteratively correct the error. Let $B_0=f^{-1}([-1,-1/3])$ and $C_0=f^{-1}([1/3,1])$, which are closed in $A$ and hence in $X$. By Urysohn's Lemma, there exists a continuous function $g_0:X\to [-1/3,1/3]$ with $g_0(B_0)=\{-1/3\}$ and $g_0(C_0)=\{1/3\}$. The function $f_1=f-g_0{|}_A$ has its range shrunk to $[-2/3,2/3]$. This process is repeated, yielding a series $g_n$ whose sum converges uniformly to the desired continuous extension $F$.

This theorem is profoundly useful. In calculus, you might extend a function from a discrete set; Tietze lets you do it from any closed set in a normal space, provided you already have a continuous starting point.

Applications and Deeper Implications

The power of these theorems is best seen in their applications. They are critical in proving major embedding theorems, such as the Urysohn Metrization Theorem, which states that every regular, second-countable space is metrizable. The proof uses Urysohn's Lemma to construct a countable family of continuous functions that, together, embed the space into the metrizable Hilbert cube $[0,1{]}^{\omega }$.

In functional analysis, Tietze's Theorem guarantees that continuous functions on a closed subset of a normal space (like a compact Hausdorff space) can be extended, which is vital for the theory of $C(X)$, the algebra of continuous real-valued functions. It tells us that closed subsets are not obstructions to extending continuous functions globally, provided the ambient space is normal. This is why normality is precisely the right condition: weaker axioms ($T_1$, Hausdorff) are insufficient, as there are Hausdorff spaces where Urysohn's Lemma fails.

Common Pitfalls

1. Assuming all Hausdorff spaces are normal: This is a frequent error. Normality is strictly stronger than the Hausdorff condition. A classic counterexample is the Moore plane (or Niemytzki plane). It is a Hausdorff, even Tychonoff, space that is not normal. Assuming normality where it doesn't hold will lead to incorrect application of both theorems.
2. Overlooking the "closed subset" condition: Both theorems have critical closedness assumptions. Urysohn's Lemma requires the sets $A$ and $B$ to be closed. Tietze's Theorem requires the subset $A$ from which you are extending to be closed in $X$. Attempting to apply them to non-closed sets (like open intervals in $\mathbb{R}$) generally fails.
3. Confusing construction with existence: Urysohn's Lemma is a constructive existence proof, but the construction is highly non-unique. The function $f$ it produces depends on the arbitrary choices of open sets made during the proof. There is not "the" Urysohn function for two sets, but rather many possible functions.
4. Misapplying the boundedness clause in Tietze: The theorem assures an extension exists. For a bounded function $f:A\to [-M,M]$, it further assures an extension $F:X\to [-M,M]$ exists. However, for an unbounded function $f:A\to \mathbb{R}$, the extension $F:X\to \mathbb{R}$ is guaranteed to exist but will typically be unbounded on $X$ as well. You cannot freely impose new bounds on the extension.

Summary

• Urysohn's Lemma converts the topological separation of disjoint closed sets in a normal space into a continuous, real-valued function that is 0 on one set and 1 on the other.
• The Tietze Extension Theorem leverages normality to continuously extend any real-valued continuous function from a closed subset of a space to the entire space, preserving bounds if present.
• Normality ($T_1$ + separation of disjoint closed sets by open neighborhoods) is the precise and necessary condition enabling these results; weaker separation axioms are insufficient.
• These theorems are not merely abstract results but are fundamental tools for proving deeper theorems in topology, such as metrization and embedding theorems, and for working with function spaces in analysis.
• Successful application requires vigilant attention to hypotheses: the sets involved must be closed, and the ambient space must be verified as normal, not just Hausdorff.