Gluing lemma for closed subsets

Statement

Let $A$ and $B$ be closed subsets of a topological space $X$ whose union is $X$, 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 $X$ 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 which cover $X$. 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

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