Interval (mathematics)

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Because an interval is also a Neighborhood of every real number ''x'' lying inside the interval, the term interval is sometimes loosely used as a synonym for '''neighborhood''' when discussing multi-dimensional metric spaces, such as the complex plane. HIGHER MATHEMATICS In higher $S$ of a Totally Ordered Set $T$ with the property that whenever $x$ and $y$ are in $S$ and As mentioned above, a particularly important case is when $T=\backslash mathbb\{R\}$, the set of Real Number s. Intervals of $\backslash mathbb\{R\}$ are of the following eleven different types (where $a$ and $b$ are real numbers, with ):
	# <math>	"a,b" class="copylinks" target="_blank">{Link without Title} =\{x\,\,a\leq x\leq b\}</math>
	# <math>[a,b)	\{x\,\,a\,\leq x<b\}</math>
	# <math>(a,b]	\{x\,\,a<x\leq b\}</math>
	# <math>(a,\infty)	\{x\,\,x>a\}</math>
	# <math>[a,\infty)	\{x\,\,x\geq a\}</math>
	# <math>(-\infty,b)	\{x\,\,x<b\}</math>
	# <math>(-\infty,b]	\{x\,\,x\leq b\}</math>
	# <math>	"-\infty,b" class="copylinks" target="_blank">{Link without Title} =\{x\,\,x \leq b\}\cup \{ -\infty\} </math>
	# <math>[-\infty,b)	\{x\,\,x < b\}\cup \{ -\infty\} </math>
	# <math>	"a,\infty" class="copylinks" target="_blank">{Link without Title} =\{x\,\,x \geq a\}\cup \{\infty\} </math>
	# <math>(a,\infty]	\{x\,\,x > a\}\cup \{\infty\} </math>

Intervals using the round brackets ( or ) as in the general interval (a,b) or specific examples (-1,3) and (2,4) are called open intervals and the endpoints are not included in the set. Intervals using the square brackets or as in the general interval or specific examples [-1,3 and are called '''closed intervals''' and the endpoints are included in the set. Intervals using both square and round brackets [ and ) or ( and as in the general intervals (a,b] and or specific examples [-1,3) and (2,4 are called '''half-closed intervals''' or '''half-open intervals'''.

Intervals play an important role in the theory of Integration , because they are the simplest Set s whose "size" or "measure" or "length" is easy to define (see above).
The concept of measure can then be extended to more complicated sets, leading to the Borel Measure and eventually to the Lebesgue Measure .

Intervals are precisely the Connected subsets of $\backslash mathbb\{R\}$. They are also precisely the Convex Subset s of $\backslash mathbb\{R\}$.
Since a Continuous image of a connected set is connected,
it follows that if $f:\backslash mathbb\{R\}\; ightarrow\backslash mathbb\{R\}$ is a continuous function and ''I'' is an interval, then its image $f(I)$ is also an interval.
This is one formulation of the Intermediate Value Theorem .


DYADIC INTERVALS


A special class of intervals on the real line are the ''dyadic intervals''. These are intervals of the form , where j and n are integers. (In some literature, other intervals with the same endpoints, such as and , are also considered to be dyadic intervals.) Dyadic intervals have some nice properties, such as the following:

• Every dyadic interval is contained in exactly one "parent" dyadic interval of twice the length.


• Every dyadic interval can be partitioned into two "child" dyadic intervals of half the length.


• If two dyadic intervals overlap, then one of them must be a subset of the other.

The dyadic intervals thus have a structure very similar to an infinite Binary Tree .

Dyadic intervals are often used in harmonic analysis, for instance to build the Haar Wavelet system.


INTERVALS IN ORDER THEORY


In Order Theory , the concept of an interval can be extended to . For example, given a partially ordered set (''P'', ≤) and two elements ''a'' and ''b'' of ''P'', one defines the set
  :T · S { ''x'' there is some ''y'' in ''T'', and some ''z'' in ''S'', such that ''x'' = ''y'' · ''z'' }
  <nowiki>]</nowiki>''a'',''b''<nowiki>[</nowiki> { ''x'' ''a'' < ''x'' < ''b'' }
  "''a'',''b''" class="copylinks" target="_blank">{Link without Title} = { ''x'' ''a'' ≤ ''x'' ≤ ''b'' }
  <nowiki>[</nowiki>''a'',''b''<nowiki>[</nowiki> { ''x'' ''a'' ≤ ''x'' < ''b'' }
  <nowiki>]</nowiki>''a'',''b''<nowiki>]</nowiki> { ''x'' ''a'' < ''x'' ≤ ''b'' }