Closed unit interval

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Definition

As a subset of the real numbers

The closed unit interval is defined as the interval $[0,1]$ or the set $\{x\in \mathbb {R} \mid 0\leq x\leq 1\}$.

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 $a\in \mathbb {R}$	equivalent as a metric space; in fact, equivalent as a subset of the metric space $\mathbb {R}$, in the sense that an isometry of $\mathbb {R}$ (translation) sends $[0,1]$ to $[a,a+1]$
$[a,b]$ for $a,b\in \mathbb {R}$, $a<b$	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