Unit interval

In mathematics, the unit interval is the interval , that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. The unit interval plays a fundamental role in homotopy theory, a major branch of topology. It is a metric space, compact, contractible, path connected and locally path connected. As a topological space, it is homeomorphic to the extended real number line. The unit interval is a one-dimensional analytical manifold with boundary {0,1}, carrying a standard orientation from 0 to 1. As a subset of the real numbers, its Lebesgue measure is 1. It is a totally ordered set and a complete lattice (every subset of the unit interval has a supremum and an infimum).

Related Topics:
Mathematics - Interval - Real number - Homotopy theory - Topology - Metric space - Compact - Contractible - Path connected - Topological space - Homeomorphic - Extended real number line - One-dimensional - Manifold - Orientation - Lebesgue measure - Totally ordered set - Complete lattice - Supremum - Infimum

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In the literature, the term "unit interval" is also sometimes applied to the other shapes that an interval from 0 to 1 could take, that is (0,1], .

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Sometimes, the term "unit interval" is used to refer to objects that play a role in various branches of mathematics analogous to the role that plays in homotopy theory. For example, in the theory of quivers, the (analogue of the) unit interval is the graph whose vertex set is {0,1} and which contains a single edge e whose source is 0 and whose target is 1. One can then define a notion of homotopy between quiver homomorphisms analogous to the notion of homotopy between continuous maps.

Related Topics:
Quiver - Homotopy - Homomorphism - Continuous

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In all of its guises, the unit interval is almost always written I, and the following ASCII picture suffices in almost any context:

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*-->--*

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0     1

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   I

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