# Gluing lemma for closed subsets

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

## Statement

Let and be closed subsets of a topological space , and and be continuous maps such that . Then there exists a unique continuous map from to whose restriction to is and to is .

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.

## 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 , 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 .

## 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.