Gluing lemma for closed subsets

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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

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.