glueing lemma for a set-theoretic map

Let \(X,Y\) be sets, \(U_i\) a set-theoretic cover and \(f_i: U_i \rightarrow Y\) be a collection of maps, such that

\begin{align*} f_{i, \vert U_i \cap U_j} = f_{j, \vert U_i \cap U_j} \end{align*}

Then there exists a map \(f: X \rightarrow Y\) such that

\begin{align*} f_{\vert U_i} = f_i && \forall i \end{align*}

Let

\begin{align*} f_i \subseteq U_i \times Y \end{align*}

Then for \(i \in I\) there exists the inclusion

\begin{align*} U_i \times Y \rightarrowtail X \times Y \end{align*}

Thus consider

\begin{align*} f \coloneqq \bigcup_{i \in I} \iota_i[f_i] \end{align*}

This relation is

1. linkstotal, since each element \(x \in X\) is an element of an \(U_i\)
2. rechtseindeutig, since the maps agree on their intersection

Furthermore, by construction,

\begin{align*} f_{\vert U_i} = f_i \end{align*}