Gluing lemma for closed subsets
This article is about the statement of a simple but indispensable lemma in topology
The result can be modified to handle finitely many closed sets which cover ; however, it does not cater to arbitrarily many closed sets. This is in contrast with the gluing lemma for open subsets.
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
Given: A topological space , closed subsets of . Continuous functions , such that .
To prove: There is a unique continuous map whose restriction to equals and whose restriction to equals .
Proof: Note that since is closed in , they are also closed in .
- There is a unique function on whose restriction to is and to is : This is set-theoretically obvious.
- This function is continuous: For this, we prove that the inverse image of any closed subset of is closed in . Let be a closed subset of . iff or . Thus, . Since is continuous, is a closed subset of , which is closed in . So is closed in . Similarly, is closed in . Since a finite union of closed subsets is closed, is closed in .
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.