In mathematics, the unit interval is the closed interval [0,1], that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted I. In addition to its role in real analysis, the unit interval is used to study homotopy theory in the field of topology.
In the literature, the term "unit interval" is sometimes applied to the other shapes that an interval from 0 to 1 could take: (0,1], [0,1), and (0,1) (see Notations for intervals). However, the notation I is most commonly reserved for the closed interval [0,1].
Properties
The unit interval is a complete metric space, homeomorphic to the extended real number line. As a topological space it is compact, contractible, path connected and locally path connected. The Hilbert cube is obtained by taking a topological product of countably many copies of the unit interval.
In mathematical analysis, the unit interval is a one-dimensional analytical manifold whose boundary consists of the two points 0 and 1. Its standard orientation goes from 0 to 1.
The unit interval is a totally ordered set and a complete lattice (every subset of the unit interval has a supremum and an infimum).
Generalizations
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 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.
References
- Robert G. Bartle, 1964, The Elements of Real Analysis, John Wiley & Sons.
Open source encyclopedia content modification information:
This page was last modified on 13 March 2010 at 22:50.
Authorship and Review
Open source encyclopedia content provided here is not reviewed directly by MedLibrary.org. Content is sourced directly from Wikipedia and is authored by an open community of volunteers. It is not produced by or in any way affiliated with MedLibrary.org.
Usage Guidelines
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article on "Unit interval", which is available in its original form here:
http://en.wikipedia.org/w/index.php?title=Unit_interval
All material adapted used from Wikipedia is available under the terms of the GNU Free Documentation License. Wikipedia® itself is a registered trademark of the Wikimedia Foundation, Inc.