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.
Well I like to think of compactness like this because it gives me the idea of a compact set in terms of closed sets. 

A collection $\mathcal{C}$ of subsets of $X$ is said to have the finite intersection property if for every finite subcollection 

$$\{C_1,\cdots C_n\}$$ 
of $\mathcal{C}$ , $\bigcap_{i=1}^{n} C_i$ should be nonempty.

Now if in a topological space every collection $\mathcal{C}$ of closed sets in $X$ have the finite intersection property , then the space $X$ is compact.

Well instinctively you would always think of a compact set as a closed set(though we have examples where compact sets are not closed.. One lies in your backyard!!)  here is a definition involving closed
 sets.

Also you can use this definition to prove the tychonoff theorem