So much of the properties of compact sets are motivated by finite sets, to the point that thinking of compact sets as topologically finite sets may yield some deeper understanding. But finite sets have the intuitive property that every subset of a finite set is also compact(also finite), why is it that compact sets give this up?
It is easy to state that they just do and provide the example $I=(0,1)$ and $K=[0,1]$ as an example, but the problem with that is it really only helps illuminate how compact sets work in the euclidean spaces. It isn't generally true in all spaces that open sets aren't compact, or that closed and bounded sets are compact. So there is a problem generalizing the ideas.
The heart of my question is: Given a subset $E$ of a compact set $K$ why isn't it compact?
Simple Answer: Because there is an open cover of $E$ which has no finite subcover
The deeper question: Why?
Best Answer
There's a nice symmetry/duality here stemming from the fact that "finite" has two generalizations in the topological category.
Images and subsets of finite sets are finite.
Discrete topological spaces generalize finite sets in that subspaces inherit the property.
Compact topological spaces generalize finite sets in that images inherit the property.
A topological space that is discrete and compact is finite.