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In [[mathematics]], the '''unit interval''' is the [[interval (mathematics)|interval]] [0,1], that is the set of all [[real number]]s <var>x</var> such that 0&nbsp;&le;&nbsp;<var>x</var>&nbsp;&le;&nbsp;1.  The unit interval plays a fundamental role in [[homotopy theory]], a major branch of [[topology]]. It is a [[metric space]], [[compact space|compact]], [[contractible]], [[connectedness|path connected]] and [[connectedness|locally path connected]]. As a [[topological space]], it is [[homeomorphism|homeomorphic]] to the [[extended real number line]]. The unit interval is a [[dimension|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 [[total order|totally ordered set]] and a complete [[lattice]] (every subset of the unit interval has a [[supremum]] and an [[infimum]]).

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], [0,1), and (0,1).  However, it's most commonly reserved for the closed interval [0,1], and Wikipedia follows this convention.

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 [0,1] plays in homotopy theory.  For example, in the theory of [[quiver (mathematics)|quiver]]s, the (analogue of the) unit interval is the graph whose vertex set is {0,1} and which contains a single edge <var>e</var> whose source is 0 and whose target is 1.  One can then define a notion of [[homotopy]] between quiver [[homomorphism]]s analogous to the notion of homotopy between [[continuous]] maps.

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

<tt>*--&gt;--*</tt><br>
<tt>0&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;1</tt><br>
<tt>&nbsp;&nbsp;&nbsp;<var>I</var></tt>

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In [[telecommunication]]s, a '''unit interval''' is defined as: In isochronous transmission, the longest interval of which the theoretical durations of the significant intervals of a signal are all whole multiples. 

Source: [[Federal Standard 1037C]] in support of [[MIL-STD-188]]

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====References====
* Adapted from the [[Wikipedia]] article, "Unit_interval" [http://en.wikipedia.org/wiki/Unit_interval http://en.wikipedia.org/wiki/Unit_interval], used under the [[GNU Free Documentation License]]