I want to calculate the Euler Characteristic of a Hemisphere with a border, without a border the Hemisphere would be homeomorphic to a Disk, now we make the disjoint union with a flat circunference and we should be able to get the Euler Characteristic by using:
$$\chi(X \cup Y)= \chi(X)+\chi(Y)$$
but somehow this doesn't feel right.
I haven't gotten to Gauss-Bonet Theorem yet, and would like to find it without it.
Thanks in advance.