# compact space

Roughly, a space that contains all of its boundary points and limit points. So, [1,0] is a compact set, but (0,1) is not a compact set.

There are notions of compactness for euclidean space, metric spaces, and topological spaces.

## 1. formal definition

- A set of points \(S\) is compact if every open cover (possibly consisting of uncountably infinite elements) has a finite subcover, that is a subset that covers \(S\)

## 2. euclidean space

- A subset of euclidean space is compact if it is a closed set and a bounded set.

## 3. infinite dimensional spaces

- for infinite dimensional spaces, we additionally need the space to be
*totally*bounded – roughly, it can ve finitely covered by a set of arbitrarily small sets