glueing lemma for open maps on an open covering

Let \(X,Y\) be topological spaces, \((U_i)\) a collection of subsets \(U_i \subseteq X\) such that \(\mathrm{int}(U_i)\) is an open cover of \(X\). Suppose \(f_i: U_i \rightarrow Y\) is a family of open maps such that \(f_{i, U_i \cap U_j} = f_{j, U_i \cap U_j}\).

Then the set-theoretic map

\begin{align*} f: X \rightarrow Y \end{align*}

induced by glueing lemma for a set-theoretic map is also open

Suppose \(O \subseteq X\) is open, then \(\mathrm{int}(U_i) \cap O \subseteq U_i\) is open. Therefore, \(f_i[ \mathrm{int}(U_i) \cap O] \subseteq Y\) is open.

Thus

\begin{align*} f[O] = \bigcup_{i \in I} f_i[ \mathrm{int}(U_i) \cap O] \end{align*}

is open as union of open sets