Gluing lemma for closed subsets

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.
 * 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.