Editing Unit interval (section)

== Properties ==

The unit interval is a [[complete metric space]], [[homeomorphism|homeomorphic]] to the [[extended real number line]]. As a [[topological space]], it is [[compact space|compact]], [[contractible]], [[connectedness|path connected]] and [[Locally connected space|locally path connected]]. The [[Hilbert cube]] is obtained by taking a [[Product topology|topological product]] of countably many copies of the unit interval.

In [[mathematical analysis]], the unit interval is a [[dimension|one-dimensional]] analytical [[manifold]] whose boundary consists of the two points 0 and 1. Its standard [[orientability|orientation]] goes from 0 to 1.

The unit interval is a [[total order|totally ordered set]] and a [[complete lattice]] (every subset of the unit interval has a [[supremum]] and an [[infimum]]).

===Cardinality===
{{Main|Cardinality of the continuum}}

The ''size'' or ''[[cardinality]]'' of a set is the number of elements it contains.

The unit interval is a [[subset]] of the [[real number]]s <math>\mathbb{R}</math>. However, it has the same size as the whole set: the [[cardinality of the continuum]]. Since the real numbers can be used to represent points along an [[Real line|infinitely long line]], this implies that a [[line segment]] of length 1, which is a part of that line, has the same number of points as the whole line. Moreover, it has the same number of points as a square of [[area]] 1, as a [[cube]] of [[volume]] 1, and even as an unbounded ''n''-dimensional [[Euclidean space]] <math>\mathbb{R}^n</math> (see [[Space filling curve]]).

The number of elements (either real numbers or points) in all the above-mentioned sets is [[Uncountable set|uncountable]], as it is strictly greater than the number of [[natural number]]s.

===Orientation===
The unit interval is a [[curve]]. The open interval (0,1) is a subset of the [[positive real numbers]] and inherits an orientation from them. The [[curve orientation|orientation]] is reversed when the interval is entered from 1, such as in the integral <math>\int_1^x \frac{dt}{t} </math> used to define [[natural logarithm]] for ''x'' in the interval, thus yielding negative values for logarithm of such ''x''. In fact, this integral is evaluated as a [[signed area]] yielding ''negative area'' over the unit interval due to reversed orientation there.