Show that if $S$ is a surface with boundary obtained by removing the interiors of $n$ disjoint closed discs from a closed surface $T$, then
(i) $\chi(S)=\chi(T)-n$
(ii) $S$ is orientable if and only if $T$ is orientable
I have just started a topology and geometry course and would greatly appreciate help with the question above. I believe this question may be using the Euler characteristic to solve it but I am unsure how to proceed. Thank you.
1) Triangulate $S$ in the following way : every disk $D$ will be triangulated with a single triangle, and triangulate the rest as you like. Now let's count : for every open disk removed, we simply lose 1 triangle, but we keep the same number of vertices/edges as they are on $\partial D$, i.e our Euler characteristic drops by 1 for every disk removed.
2) $S$ is orientable if and only if $S$ does contains a Moebius band : and removing or adding a disk does not changes the fact that $S$ contains a Moebius band.