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A point on the boundary of a domain together with the class of equivalent paths leading from the interior of the domain to that point. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013900/a0139001.png" /> be a point on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013900/a0139002.png" /> of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013900/a0139003.png" /> in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013900/a0139004.png" />-plane and let there exist a path described by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013900/a0139005.png" />, where the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013900/a0139006.png" /> is defined and continuous on a certain segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013900/a0139007.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013900/a0139008.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013900/a0139009.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013900/a01390010.png" />. One then says that this path leads to the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013900/a01390011.png" /> (from the inside of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013900/a01390012.png" />) and defines the attainable boundary point represented by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013900/a01390013.png" />. Two paths leading to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013900/a01390014.png" /> are said to be equivalent (or, defining the same attainable boundary point) if there exists a third path which also leads to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013900/a01390015.png" /> from the inside of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013900/a01390016.png" /> and which has non-empty intersections inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013900/a01390017.png" /> as close to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013900/a01390018.png" /> as one pleases with each of the two paths considered. The totality of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013900/a01390019.png" /> and the class of equivalent paths leading to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013900/a01390020.png" /> from the interior of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013900/a01390021.png" /> is said to be an attainable boundary point of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013900/a01390022.png" />. Not every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013900/a01390023.png" /> represents an attainable boundary point; on the other hand, the same point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013900/a01390024.png" /> can represent several, or even an infinite set of different, attainable boundary points.
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A point on the boundary of a domain together with the class of equivalent paths leading from the interior of the domain to that point. Let $  \xi $
 +
be a point on the boundary $  \partial  G $
 +
of a domain $  G $
 +
in the complex $  z $-
 +
plane and let there exist a path described by the equation $  z = z (t) $,  
 +
where the function $  z (t) $
 +
is defined and continuous on a certain segment $  [ \alpha , \beta ] $,  
 +
$  z (t) \in G $
 +
if $  \alpha \leq  t < \beta $,  
 +
$  z ( \beta ) = \xi $.  
 +
One then says that this path leads to the point $  \xi $(
 +
from the inside of $  G $)  
 +
and defines the attainable boundary point represented by $  \xi $.  
 +
Two paths leading to $  \xi $
 +
are said to be equivalent (or, defining the same attainable boundary point) if there exists a third path which also leads to $  \xi $
 +
from the inside of $  G $
 +
and which has non-empty intersections inside $  G $
 +
as close to $  \xi $
 +
as one pleases with each of the two paths considered. The totality of a point $  \xi \in \partial  G $
 +
and the class of equivalent paths leading to $  \xi $
 +
from the interior of $  G $
 +
is said to be an attainable boundary point of the domain $  G $.  
 +
Not every point $  \xi \in \partial  G $
 +
represents an attainable boundary point; on the other hand, the same point $  \xi \in \partial  G $
 +
can represent several, or even an infinite set of different, attainable boundary points.
  
 
An attainable boundary point is the unique point of a prime end (cf. [[Limit elements|Limit elements]]) of the first kind; a (multi-point) prime end of the second kind contains exactly one attainable boundary point, while prime ends of the third and fourth kinds do not contain attainable boundary points. Each point of the boundary of a Jordan domain is attainable.
 
An attainable boundary point is the unique point of a prime end (cf. [[Limit elements|Limit elements]]) of the first kind; a (multi-point) prime end of the second kind contains exactly one attainable boundary point, while prime ends of the third and fourth kinds do not contain attainable boundary points. Each point of the boundary of a Jordan domain is attainable.
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''3''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.F. Collingwood,  A.J. Lohwater,  "The theory of cluster sets" , Cambridge Univ. Press  (1966)  pp. Chapt. 1;6</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''3''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.F. Collingwood,  A.J. Lohwater,  "The theory of cluster sets" , Cambridge Univ. Press  (1966)  pp. Chapt. 1;6</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
The standard Western terminology is accessible boundary point.
 
The standard Western terminology is accessible boundary point.

Latest revision as of 18:48, 5 April 2020


A point on the boundary of a domain together with the class of equivalent paths leading from the interior of the domain to that point. Let $ \xi $ be a point on the boundary $ \partial G $ of a domain $ G $ in the complex $ z $- plane and let there exist a path described by the equation $ z = z (t) $, where the function $ z (t) $ is defined and continuous on a certain segment $ [ \alpha , \beta ] $, $ z (t) \in G $ if $ \alpha \leq t < \beta $, $ z ( \beta ) = \xi $. One then says that this path leads to the point $ \xi $( from the inside of $ G $) and defines the attainable boundary point represented by $ \xi $. Two paths leading to $ \xi $ are said to be equivalent (or, defining the same attainable boundary point) if there exists a third path which also leads to $ \xi $ from the inside of $ G $ and which has non-empty intersections inside $ G $ as close to $ \xi $ as one pleases with each of the two paths considered. The totality of a point $ \xi \in \partial G $ and the class of equivalent paths leading to $ \xi $ from the interior of $ G $ is said to be an attainable boundary point of the domain $ G $. Not every point $ \xi \in \partial G $ represents an attainable boundary point; on the other hand, the same point $ \xi \in \partial G $ can represent several, or even an infinite set of different, attainable boundary points.

An attainable boundary point is the unique point of a prime end (cf. Limit elements) of the first kind; a (multi-point) prime end of the second kind contains exactly one attainable boundary point, while prime ends of the third and fourth kinds do not contain attainable boundary points. Each point of the boundary of a Jordan domain is attainable.

References

[1] A.I. Markushevich, "Theory of functions of a complex variable" , 3 , Chelsea (1977) (Translated from Russian)
[2] E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets" , Cambridge Univ. Press (1966) pp. Chapt. 1;6

Comments

The standard Western terminology is accessible boundary point.

How to Cite This Entry:
Attainable boundary point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Attainable_boundary_point&oldid=14913
This article was adapted from an original article by E.P. Dolzhenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article