Motivation for the Definition of Compact Space A compact topological space is defined as a space, $C$, such that for any set $\mathcal{A}$ of open sets such that $C \subseteq \bigcup_{U\in \mathcal{A}} U$, there is finite set $\mathcal{A'} \subseteq \mathcal{A}$ such that $C \subseteq \bigcup_{U'\in \mathcal{A'}} U'$.
Now, this definition leads to many interesting results, but if I were teaching someone about compact sets, how would I motivate this? Concepts like sequential compactness, open and closedness, and even connectedness are reasonably easy to motivate. I can not see how to motivate this definition. Compact spaces are often seen as generalizations of finite spaces. They are also seen as a generalization of boundedness and closedness. I can't see how to connect the definition with these concepts.
Alternatively, is there a definition of a compact set which is easier to motivate?

According to Munkres, the original definition of compactness is a space which satisfies the Bolzano-Weierstrauss property holds.  That is, if every infinite subset has a limit-point.  
Unfortunately, it turns out, this conception of compactness,sometimes called limit point compactness, doesn't have all the useful properties that compactness has.
For example, the continuous image of a limit point compact space need not be limit point compact.  Also, a limit point compact subspace of a Hausdorff space need not be closed.
