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A non-empty connected open set in a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d0337601.png" />. The closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d0337602.png" /> of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d0337603.png" /> is called a closed domain; the closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d0337604.png" /> is called the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d0337605.png" />. The points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d0337606.png" /> are also called the interior points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d0337607.png" />; the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d0337608.png" /> are called the boundary points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d0337609.png" />; the points of the complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376010.png" /> are called the exterior points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376011.png" />.
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Any two points of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376012.png" /> in the real Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376014.png" /> (or in the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376016.png" />, or on a Riemann surface or in a Riemannian domain), can be joined by a path (or arc) lying completely in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376017.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376018.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376019.png" />, they can even be joined by a polygonal path with a finite number of edges. Finite and infinite open intervals are the only domains in the real line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376020.png" />; their boundaries consist of at most two points. A domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376021.png" /> in the plane is called simply connected if any closed path in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376022.png" /> can be continuously deformed to a point, remaining throughout in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376023.png" />. In general, the boundary of a simply-connected domain in the (open) plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376024.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376025.png" /> can consist of any number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376026.png" /> of connected components, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376027.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376028.png" /> is regarded as a domain in the compact extended plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376029.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376030.png" /> and the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376031.png" /> of boundary components is finite, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376032.png" /> is called the connectivity order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376033.png" />; for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376035.png" /> is called multiply connected. In other words, the connectivity order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376036.png" /> is one more than the minimum number of cross-cuts joining components of the boundary in pairs that are necessary to make <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376037.png" /> simply connected. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376039.png" /> is called doubly connected, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376040.png" />, triply connected, etc.; for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376041.png" /> one has finitely-connected domains and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376042.png" /> infinitely-connected domains. The connectivity order of a plane domain characterizes its topological type. The topological types of domains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376044.png" />, or in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376046.png" />, cannot be characterized by a single number.
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Even for a simply-connected plane domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376047.png" /> the metric structure of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376048.png" /> can be very complicated (see [[Limit elements|Limit elements]]). In particular, the boundary points can be divided into accessible points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376049.png" />, for which there exists a path <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376053.png" />, joining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376054.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376055.png" /> with any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376056.png" />, and inaccessible points, for which no such paths exists (cf. [[Attainable boundary point|Attainable boundary point]]). For any simply-connected plane domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376057.png" /> the set of accessible points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376058.png" /> is everywhere dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376059.png" />.
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A non-empty connected open set in a topological space  $  X $.
 +
The closure  $  \overline{D}\; $
 +
of a domain $  D $
 +
is called a closed domain; the closed set  $  \textrm{ Fr }  D = \overline{D}\; \setminus  D $
 +
is called the boundary of  $  D $.  
 +
The points  $  x \in D $
 +
are also called the interior points of $  D $;
 +
the points  $  x \in \textrm{ Fr }  D $
 +
are called the boundary points of  $  D $;
 +
the points of the complement  $  C \overline{D}\; = X \setminus  \overline{D}\; $
 +
are called the exterior points of $  D $.
  
A domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376060.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376061.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376062.png" /> is called bounded, or finite, if
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Any two points of a domain  $  D $
 +
in the real Euclidean space  $  \mathbf R  ^ {n} $,
 +
$  n \geq  1 $(
 +
or in the complex space  $  \mathbf C  ^ {m} $,
 +
$  m \geq  1 $,
 +
or on a Riemann surface or in a Riemannian domain), can be joined by a path (or arc) lying completely in  $  D $;
 +
if  $  D \subset  \mathbf R  ^ {n} $
 +
or  $  D \subset  \mathbf C  ^ {m} $,
 +
they can even be joined by a polygonal path with a finite number of edges. Finite and infinite open intervals are the only domains in the real line  $  \mathbf R = \mathbf R  ^ {1} $;
 +
their boundaries consist of at most two points. A domain  $  D $
 +
in the plane is called simply connected if any closed path in  $  D $
 +
can be continuously deformed to a point, remaining throughout in  $  D $.  
 +
In general, the boundary of a simply-connected domain in the (open) plane  $  \mathbf R  ^ {2} $
 +
or  $  \mathbf C = \mathbf C  ^ {1} $
 +
can consist of any number  $  k $
 +
of connected components,  $  0 \leq  k \leq  \infty $.  
 +
If  $  D $
 +
is regarded as a domain in the compact extended plane  $  \overline{\mathbf R}\; {}  ^ {2} $
 +
or  $  \overline{\mathbf C}\; $
 +
and the number  $  k $
 +
of boundary components is finite, then  $  k $
 +
is called the connectivity order of  $  D $;
 +
for  $  k > 1 $,
 +
$  D $
 +
is called multiply connected. In other words, the connectivity order  $  k $
 +
is one more than the minimum number of cross-cuts joining components of the boundary in pairs that are necessary to make  $  D $
 +
simply connected. For  $  k = 2 $,
 +
$  D $
 +
is called doubly connected, for  $  k = 3 $,
 +
triply connected, etc.; for  $  k < \infty $
 +
one has finitely-connected domains and for  $  k = \infty $
 +
infinitely-connected domains. The connectivity order of a plane domain characterizes its topological type. The topological types of domains in  $  \mathbf R  ^ {n} $,
 +
$  n \geq  3 $,  
 +
or in  $  \mathbf C  ^ {m} $,
 +
$  m \geq  2 $,  
 +
cannot be characterized by a single number.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376063.png" /></td> </tr></table>
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Even for a simply-connected plane domain  $  D $
 +
the metric structure of the boundary  $  \textrm{ Fr }  D $
 +
can be very complicated (see [[Limit elements|Limit elements]]). In particular, the boundary points can be divided into accessible points  $  x _ {0} \in \textrm{ Fr }  D $,
 +
for which there exists a path  $  x ( t) $,
 +
$  0 \leq  t \leq  1 $,
 +
$  x ( 0) \in D $,
 +
$  x ( 1) = x _ {0} $,
 +
joining  $  x _ {0} $
 +
in  $  D $
 +
with any point  $  x ( 0) \in D $,
 +
and inaccessible points, for which no such paths exists (cf. [[Attainable boundary point|Attainable boundary point]]). For any simply-connected plane domain  $  D $
 +
the set of accessible points of  $  \textrm{ Fr }  D $
 +
is everywhere dense in  $  \textrm{ Fr }  D $.
  
if not, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376064.png" /> is called unbounded or infinite. A closed plane Jordan curve divides the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376065.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376066.png" /> into two Jordan domains: A finite domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376067.png" /> and an infinite domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376068.png" />. All boundary points of a Jordan domain are accessible.
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A domain  $  D $
 +
in  $  \mathbf R  ^ {n} $
 +
or  $  \mathbf C  ^ {m} $
 +
is called bounded, or finite, if
  
 +
$$
 +
\sup \
 +
\{ {| x | } : {
 +
x \in D } \}
 +
<  \infty ;
 +
$$
  
 +
if not,  $  D $
 +
is called unbounded or infinite. A closed plane Jordan curve divides the plane  $  \mathbf R  ^ {2} $
 +
or  $  \mathbf C $
 +
into two Jordan domains: A finite domain  $  D  ^ {+} $
 +
and an infinite domain  $  D  ^ {-} $.
 +
All boundary points of a Jordan domain are accessible.
  
 
====Comments====
 
====Comments====
Instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376069.png" />, the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376070.png" /> is also denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376071.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033760/d03376072.png" />.
+
Instead of $  \textrm{ Fr }  D $,  
 +
the boundary of $  D $
 +
is also denoted by $  \textrm{ b } D $
 +
or $  \partial  D $.
  
 
From the definition it can be seen that a domain is bounded if (and only if) it is contained in a ball centred at the coordinate origin and of finite radius.
 
From the definition it can be seen that a domain is bounded if (and only if) it is contained in a ball centred at the coordinate origin and of finite radius.

Latest revision as of 19:36, 5 June 2020


A non-empty connected open set in a topological space $ X $. The closure $ \overline{D}\; $ of a domain $ D $ is called a closed domain; the closed set $ \textrm{ Fr } D = \overline{D}\; \setminus D $ is called the boundary of $ D $. The points $ x \in D $ are also called the interior points of $ D $; the points $ x \in \textrm{ Fr } D $ are called the boundary points of $ D $; the points of the complement $ C \overline{D}\; = X \setminus \overline{D}\; $ are called the exterior points of $ D $.

Any two points of a domain $ D $ in the real Euclidean space $ \mathbf R ^ {n} $, $ n \geq 1 $( or in the complex space $ \mathbf C ^ {m} $, $ m \geq 1 $, or on a Riemann surface or in a Riemannian domain), can be joined by a path (or arc) lying completely in $ D $; if $ D \subset \mathbf R ^ {n} $ or $ D \subset \mathbf C ^ {m} $, they can even be joined by a polygonal path with a finite number of edges. Finite and infinite open intervals are the only domains in the real line $ \mathbf R = \mathbf R ^ {1} $; their boundaries consist of at most two points. A domain $ D $ in the plane is called simply connected if any closed path in $ D $ can be continuously deformed to a point, remaining throughout in $ D $. In general, the boundary of a simply-connected domain in the (open) plane $ \mathbf R ^ {2} $ or $ \mathbf C = \mathbf C ^ {1} $ can consist of any number $ k $ of connected components, $ 0 \leq k \leq \infty $. If $ D $ is regarded as a domain in the compact extended plane $ \overline{\mathbf R}\; {} ^ {2} $ or $ \overline{\mathbf C}\; $ and the number $ k $ of boundary components is finite, then $ k $ is called the connectivity order of $ D $; for $ k > 1 $, $ D $ is called multiply connected. In other words, the connectivity order $ k $ is one more than the minimum number of cross-cuts joining components of the boundary in pairs that are necessary to make $ D $ simply connected. For $ k = 2 $, $ D $ is called doubly connected, for $ k = 3 $, triply connected, etc.; for $ k < \infty $ one has finitely-connected domains and for $ k = \infty $ infinitely-connected domains. The connectivity order of a plane domain characterizes its topological type. The topological types of domains in $ \mathbf R ^ {n} $, $ n \geq 3 $, or in $ \mathbf C ^ {m} $, $ m \geq 2 $, cannot be characterized by a single number.

Even for a simply-connected plane domain $ D $ the metric structure of the boundary $ \textrm{ Fr } D $ can be very complicated (see Limit elements). In particular, the boundary points can be divided into accessible points $ x _ {0} \in \textrm{ Fr } D $, for which there exists a path $ x ( t) $, $ 0 \leq t \leq 1 $, $ x ( 0) \in D $, $ x ( 1) = x _ {0} $, joining $ x _ {0} $ in $ D $ with any point $ x ( 0) \in D $, and inaccessible points, for which no such paths exists (cf. Attainable boundary point). For any simply-connected plane domain $ D $ the set of accessible points of $ \textrm{ Fr } D $ is everywhere dense in $ \textrm{ Fr } D $.

A domain $ D $ in $ \mathbf R ^ {n} $ or $ \mathbf C ^ {m} $ is called bounded, or finite, if

$$ \sup \ \{ {| x | } : { x \in D } \} < \infty ; $$

if not, $ D $ is called unbounded or infinite. A closed plane Jordan curve divides the plane $ \mathbf R ^ {2} $ or $ \mathbf C $ into two Jordan domains: A finite domain $ D ^ {+} $ and an infinite domain $ D ^ {-} $. All boundary points of a Jordan domain are accessible.

Comments

Instead of $ \textrm{ Fr } D $, the boundary of $ D $ is also denoted by $ \textrm{ b } D $ or $ \partial D $.

From the definition it can be seen that a domain is bounded if (and only if) it is contained in a ball centred at the coordinate origin and of finite radius.

How to Cite This Entry:
Domain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Domain&oldid=17271
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article