Closed unit interval

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This article is about a particular topological space (uniquely determined up to homeomorphism)|View a complete list of particular topological spaces

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