Adding A and B creates a new set C. In order for C to be compact, it must have its own set of open covers where the union of a certain collection of them (much like the union of open covers that contain A and B above) where $\displaystyle C \subseteq \cup Q_{\gamma}$. That means:

$\displaystyle C = (A+B) \subseteq ( \cup O_{\alpha} + \cup P_{\beta})$

then since $\displaystyle C = A+B \Rightarrow C \subseteq (\cup O_{\alpha} + \cup P_{\beta})$

also since $\displaystyle C = A+B \Rightarrow A+B \subseteq (\cup Q_{\gamma})$

which implies $\displaystyle \cup O_{\alpha} + \cup P_{\beta} = \cup Q_{\gamma}$ so c is compact.

Input please! Thanks again in advance.