# Closed unit interval

From Topospaces

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## Contents

## Definition

### As a subset of the real numbers

The **closed unit interval** is defined as the interval or the set . It can also be defined as the closed disk with center and radius , i.e., the set:

### As a metric space

The **closed unit interval** is the metric space with the Euclidean metric.

### As a manifold with boundary

*Fill this in later*

### As a topological space

The **closed unit interval** is the set with the subspace topology induced from the real line.

## Equivalent spaces

Space | How strongly is it equivalent to the closed unit interval? |
---|---|

for | equivalent as a metric space; in fact, equivalent as a subset of the metric space , in the sense that an isometry of (translation) sends to |

for , | 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 and | 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 |