Homology and topological propeties i have this theorem with it's proof but i don't understand the last part 
They use this proposition:

My question is Why $\varphi^c\cap U_i$ is closed and pairwise disjoint ? where $\varphi^c=\lbrace x, \varphi(x)\leq c\rbrace$
Please thank you.

The theorem is applied (see the middle in $6.6$ ) to $X=\varphi^c\cap C$ and $X_i=\varphi^c\cap U_i$ which is a closed subset in $X$.($X_i=X\cap(\cup_{j\neq i}U_j)^c$ where the c's power is the complementary ).
