Compactness
Compact sets can be thought of as a generalization of [what?]. See the reverse side for a justification.
There are multiple ways of formulating compactness, with some being more intuitive than others.
There are 3 formulations that are particularly important.
1. Compactness, in terms of sequences in .
A set
What is the topological generalization of this definition?
The most basic example of such a closed set in
Bolzano-Weierstrass's theorem connects the first and second formulations.
2. Compactness, in terms of [something], [something] sets.
A set
The next formulation is the most general of the three, graduating to the topic of topology. It uses the idea of a cover and a subcover.
Cover
Let
Given a cover
3. Compactness, in terms of covers.
A topological space
What is the theorem that connects these three formulations of compactness?