# closed set

## 1. definition

- A closed set is a set whose complement is an open set

## 2. for metric spaces

- For a complete metric space, a set is called closed if it is closed under the limit operation.
- What is a complete metric space? For a set \(X\) and metric \(d\), a metric space \((X,d)\) is closed, if every Cauchy sequence has a limit that exists in \(X\)
- What is a Cauchy sequence? It is a sequence $x
_{1}, x_{2},…$ such that for every \(r>0\), there exists \(N\) such that \(m,n > N\) implies that \(d(x_m,x_n) < r\). That is, all the points in the sequence eventually get close together.