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?

One of my favorite textbooks is Klaus Janich's Topology, and he has a nice motivation for compactness I feel, namely why we should care about. This is in addition to my comment about compact subsets of a Hausdorff space being essentially like finite point sets. But he writes:

In compact spaces, the following generalization from "local" to "global" properties is possible: Let $X$ be a compact space and $P$ a property that open subsets of $X$ may or may not have, and such that if $U$ and $V$ have it, then so does $U\cup V$. Then if $X$ has this property locally, i.e. every point has a neighborhood with property $P$, then $X$ itself has property $P$.

This is nice, but it is slightly advanced, and he gives some examples that follow like a continuous/locally bounded map from a compact space to $\mathbb{R}$ is bounded, and some discussions of locally finite covers and manifolds (honestly, I like this book after the fact of learning topology, not to learn from).

Hope that helps somewhat.

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.

An equivalent def'n is that if $ F$ is a non-empty family of closed sets with the F.I.P. (Finite Intersection Property) then $\cap F \not = \phi $ . This generalizes the idea of limits , and you can show that many results, e.g. on bounded closed subsets of $ R^n$ , using this property, so it is seen to be a useful tool that a space is compact. Once you show some additional consequences, e.g. that a continuous image of a compact set is compact, you can show how to apply them, e.g. in analysis, showing that an extremum exists, (hence the Mean Value Theorem in calculus). So you get easier results and new ones, from the compactness.