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 \( \mathbb{R} \).
A set \( K \subset \mathbb{R} \) is compact iff [what?].
What is the topological generalization of this definition?
The most basic example of such a closed set in \( \mathbb{R} \) is [a what?].
Bolzano-Weierstrass's theorem connects the first and second formulations.
2. Compactness, in terms of [something], [something] sets.
A set \( K \subset \mathbb{R}^n \) is compact iff it is [what and what?].
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 \( X \) be a topological space. A cover of \( X \) is a family \( (U_i)_{i \in I} \) of subsets of \( X \) such that [...]. It is finite iff the indexing set \( I \) is finite, and open iff \( U_i \) is open for each \( i \in I \).
Given a cover \( (U_i)_{i \in I} \) and \( J \subseteq I \), we say that [...] is a subcover of \( (U_i)_{i \in I} \) if it is itself a cover of \( X \).
3. Compactness, in terms of covers.
A topological space \( X \) is compact iff [what?].
What is the theorem that connects these three formulations of compactness?