Math and science::Topology

# 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?