Lemma 3.5.2 (Gluing Lemma for Continuous Functions).label Let $X,Y$ be topological spaces and $\curl{U_i}_{i\in I}$ a countable open covering of $X$. Given functions $\curl{f_i}_{i\in I}$ satisfying:

1. (1)

   For each $i\in I$, $f_{i}\in C(U_{i};Y)$

2. (2)

   For each $i,j\in I$ such that $U_{i}\cap U_{j}\neq\emp$, $f_{i}|_{U_i\cap U_j}=f_{j}|_{U_i\cap U_j}$

then there uniquely exists $f\in C(X;Y)$ such that $f|_{U_i}=f_{i}$ for all $i\in I$.