Say X is a topological space and F is in X. If C is contained in F and compact in F, then C is compact in X. This is obvious when you draw a picture, but proving it is a little more difficult. By hypothesis, C is compact in F so C has a finite subcover M, with M being the union of m_1, m_2, m_3, and so on up through m_n. Now each m is in F since C is compact in F, and F is contained in X, so each m is in X. But then M would also be contained in X since you union all the m's that are in X. So then C would have a finite subcover in X, so C is compact in X. Are there flaws with this line or reasoning? Thank you for the insight.