# 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?