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 $A\cup B$.

1. 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.
2. 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.