Gluing lemma for closed subsets

This article is about the statement of a simple but indispensable lemma in topology

Statement

Let $A$ and $B$ be closed subsets of a topological space $X$ , and $f : A \to Y$ and $g : B \to Y$ be continuous maps such that $f (x) = g (x) \forall x \in A \cap B$ . Then there exists a unique continuous map from $A \cup B$ to $Y$ whose restriction to $A$ is $f$ and to $B$ is $g$ .

The result can be modified to handle finitely many closed sets which cover $X$ ; however, it does not cater to arbitrarily many closed sets. This is in contrast with the gluing lemma for open subsets.

Related results

Gluing lemma for open subsets

Proof

The proof uses the following key facts:

A map is continuous if and only if the inverse image of any closed subset is closed
A closed subset of a closed subset is closed. For full proof, refer: Closedness is transitive
A union of two closed subsets is closed

Proof details

Given: A topological space $X$ , closed subsets $A, B$ of $X$ . Continuous functions $f, g : A \cup B \to Y$ , such that $f |_{A \cap B} = g |_{A \cap B}$ .

To prove: There is a unique continuous map $h : A \cup B \to Y$ whose restriction to $A$ equals $f$ and whose restriction to $B$ equals $g$ .

Proof: Note that since $A, B$ is closed in $X$ , they are also closed in $A \cup B$ .

There is a unique function $h$ on $A \cup B \to Y$ whose restriction to $A$ is $f$ and to $B$ is $g$ : This is set-theoretically obvious.
This function is continuous: For this, we prove that the inverse image of any closed subset of $Y$ is closed in $A \cup B$ . Let $C$ be a closed subset of $Y$ . $h (x) \in C$ iff $f (x) \in C$ or $g (x) \in C$ . Thus, $h^{- 1} (C) = f^{- 1} (C) \cup g^{- 1} (C)$ . Since $f$ is continuous, $f^{- 1} (C)$ is a closed subset of $A$ , which is closed in $A \cup B$ . So $f^{- 1} (C)$ is closed in $A \cup B$ . Similarly, $g^{- 1} (C)$ is closed in $A \cup B$ . Since a finite union of closed subsets is closed, $f^{- 1} (C) \cup g^{- 1} (C) = h^{- 1} (C)$ is closed in $A \cup B$ .

Applications

The gluing lemma for closed subsets is one of the many results in point-set topology which is applied everywhere, often without even consciously realizing it. Here are some examples:

The multiplication defined in the fundamental group and higher homotopy grooups, uses the gluing lemma (to argue that a composite of loops is a loop)
The fact that homotopies can be composed also uses the gluing lemma
Many of the proofs involving manifolds, for instance, the proof that the inclusion of a point in a manifold is a cofibration, or the proof that connected manifolds are homogeneous, uses the gluing lemma; we glue an explicit map in a neighbourhood of the point with a constant map outside.