Gluing lemma for open subsets

Statement

Let $\{U_i{\}}_{i\in I}$ be an open cover of a topological space $X$, and $f_i:U_i\to Y$ be continuous maps, such that for $x\in U_i\cap U_j$ we have $f_i(x)=f_j(x)$.

Then there exists a unique map $f:X\to Y$ such that $f{|}_{U_i}=f_i$.

This is the proof that the presheaf of continuous functions to $Y$, is actually a sheaf.

Related results

• Gluing lemma for closed subsets

Proof

The key facts used in the proof are:

• A map of topological spaces is continuous iff the inverse image of any open set is open
• An open subset of an open subset is open in the whole space
• An arbitrary union of open subsets is open