Gluing lemma for open subsets

Statement

Let ${U_{i}}_{i \in I}$ be a collection of open subsets 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 $f_{i} (x) = f_{j} (x)$ .

Let $U$ be the union of the $U_{i}$ s. Then there exists a unique map $f : U \to Y$ 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

Proof details

Given: An open cover ${U_{i}}_{i \in I}$ of a topological space $X$ . Continuous maps $f_{i} : U_{i} \to Y$ , such that for $x \in U_{i} \cap U_{j}$ , we have $f_{i} (x) = f_{j} (x)$ . $U$ is the union of the $U_{i}$ s.

To prove: There exists a unique map $f : U \to Y$ such that $f |_{U_{i}} = f_{i}$ .

Proof: Note first that the $U_{i}$ s are all open in $X$ , hence also in $U$ .

There exists a unique function $f$ on $U$ such that $f |_{U_{i}} = f_{i}$ for all $i$ : For any $x \in X$ , pick any $i$ such that $x \in U_{i}$ , and define $f (x) = f_{i} (x)$ . Such an $i$ exists because $U$ is the union of the $U_{i}$ s. Further, the definition of $f (x)$ is independent of the choice of $i$ because if $x \in U_{i} \cap U_{j}$ , $f_{i} (x) = f_{j} (x)$ . Moreover, this is the only possible way to define $f$ .
$f$ is continuous, i.e., if $V$ is an open subset of $Y$ , $f^{- 1} (V)$ is an open subset of $U$ : If $f (x) \in V$ , then $f_{i} (x) \in V$ for some $i$ . Thus, we have $f^{- 1} (V) = ⋃_{i} f_{i}^{- 1} (V)$ . Since $f_{i} : U_{i} \to Y$ is continuous, $f_{i}^{- 1} (V)$ is open in $U_{i}$ . Since open subsets of open subsets are open, and $U_{i}$ is open in $U$ , $f_{i}^{- 1} (V)$ is open in $U$ . Thus, the union $f^{- 1} (V)$ of all the $f_{i}^{- 1} (V)$ is also an open subset of $U$ .