# Closed unit interval

## Definition

### As a subset of the real numbers

The closed unit interval is defined as the interval $[0,1]$ or the set $\{ x \in \R \mid 0 \le x \le 1\}$. It can also be defined as the closed disk with center $1/2$ and radius $1/2$, i.e., the set:

$\{ x \in \R \mid |x - 1/2| \le 1/2\}$

### As a metric space

The closed unit interval is the metric space $[0,1]$ with the Euclidean metric.

### As a manifold with boundary

Fill this in later

### As a topological space

The closed unit interval is the set $[0,1]$ with the subspace topology induced from the real line.

## Equivalent spaces

Space How strongly is it equivalent to the closed unit interval?
$[a,a+1]$ for $a \in \R$ equivalent as a metric space; in fact, equivalent as a subset of the metric space $\R$, in the sense that an isometry of $\R$ (translation) sends $[0,1]$ to $[a,a + 1]$
$[a,b]$ for $a,b \in \R$, $a < b$ equivalent as a (differential) manifold with boundary and hence also as a topological space. Conformally equivalent as a metric space or as a Riemannian manifold with boundary.
Two-point compatification of real line, with points introduced at $-\infty$ and $+\infty$ equivalent as a (differential) manifold with boundary and hence also as a topological space.
Any compact 1-manifold with boundary equivalent as a (differential) manifold with boundary.
Any contractible space homotopy-equivalent