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''in potential theory''
 
''in potential theory''
  
The weakest topology in which all superharmonic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f0401901.png" /> are continuous. Objects related to the fine topology are described as  "fine" ,  "finely" , etc.
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The weakest topology in which all superharmonic functions on $  \mathbf R  ^ {n} $
 +
are continuous. Objects related to the fine topology are described as  "fine" ,  "finely" , etc.
  
The notion of fine topology is closely connected with that of a [[Thin set|thin set]] (cf. also [[Thinness of a set|Thinness of a set]]). The fine topology is stronger that the usual Euclidean topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f0401902.png" />, i.e. every Euclidean-open set is finely open. A fine neighbourhood of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f0401903.png" /> is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f0401904.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f0401905.png" /> and such that the complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f0401906.png" /> is a thin set at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f0401907.png" />. The finely-open sets are unions of pre-images under mappings by superharmonic functions of the extended real line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f0401908.png" /> and of intervals of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f0401909.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019012.png" />. Every superharmonic function on an open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019013.png" /> is finely continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019014.png" />. A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019015.png" /> is thin at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019016.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019017.png" /> is a finely-isolated point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019018.png" />.
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The notion of fine topology is closely connected with that of a [[Thin set|thin set]] (cf. also [[Thinness of a set|Thinness of a set]]). The fine topology is stronger that the usual Euclidean topology on $  \mathbf R  ^ {n} $,  
 +
i.e. every Euclidean-open set is finely open. A fine neighbourhood of a point $  x _ {0} \in \mathbf R  ^ {n} $
 +
is a set $  V ( x _ {0} ) $
 +
such that $  x _ {0} \in V ( x _ {0} ) $
 +
and such that the complement $  {C V ( x _ {0} ) } $
 +
is a thin set at $  x $.  
 +
The finely-open sets are unions of pre-images under mappings by superharmonic functions of the extended real line $  \overline{\mathbf R} $
 +
and of intervals of the form $  ( a, + \infty ] $,
 +
$  [- \infty , b) $,
 +
$  ( a, b) $,  
 +
$  - \infty < a < b < + \infty $.  
 +
Every superharmonic function on an open set $  E \subset  \mathbf R  ^ {n} $
 +
is finely continuous on $  E $.  
 +
A set $  E \subset  \mathbf R  ^ {n} $
 +
is thin at a point $  x _ {0} \in E $
 +
if and only if $  x _ {0} $
 +
is a finely-isolated point of $  E $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019019.png" /> be a fine-limit point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019020.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019021.png" /> is not thin at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019022.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019023.png" /> be a function defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019024.png" />. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019025.png" /> is called the fine limit of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019026.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019027.png" /> if for every neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019028.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019029.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019030.png" /> there exists a fine neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019031.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019032.png" /> such that
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Let $  x _ {0} $
 +
be a fine-limit point of $  E $,  
 +
that is, $  E $
 +
is not thin at $  x _ {0} $,  
 +
and let f $
 +
be a function defined on $  E $.  
 +
The number $  \lambda $
 +
is called the fine limit of f $
 +
at $  x _ {0} $
 +
if for every neighbourhood $  U ( \lambda ) $
 +
of $  \lambda $
 +
in $  \overline{\mathbf R} $
 +
there exists a fine neighbourhood $  V ( x _ {0} ) $
 +
of $  x _ {0} $
 +
such that
  
<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/f/f040/f040190/f04019033.png" /></td> </tr></table>
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$$
 +
x \in E \cap V ( x _ {0} )  \Rightarrow  f ( x) \in U ( \lambda ).
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019034.png" /> is the fine limit of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019035.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019036.png" />, then there exists a fine neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019037.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019038.png" /> is an ordinary limit at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019039.png" /> of the restriction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019040.png" /> (Cartan's theorem).
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If $  \lambda $
 +
is the fine limit of f $
 +
at $  x _ {0} $,  
 +
then there exists a fine neighbourhood $  V ( x _ {0} ) $
 +
such that $  \lambda $
 +
is an ordinary limit at $  x _ {0} $
 +
of the restriction $  f \mid  _ {E \cap V ( x _ {0}  ) } $ (Cartan's theorem).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019041.png" /> be a closed set, thin at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019042.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019043.png" /> be a superharmonic function defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019044.png" /> in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019045.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019046.png" /> has a fine limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019047.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040190/f04019048.png" />.
+
Let $  E $
 +
be a closed set, thin at a point $  x _ {0} $,  
 +
and let f > 0 $
 +
be a superharmonic function defined on $  C E $
 +
in a neighbourhood of $  x _ {0} $.  
 +
Then f $
 +
has a fine limit $  \lambda $
 +
at $  x _ {0} $.
  
 
A fine topology has also been constructed in axiomatic potential theory (cf. [[#References|[3]]]).
 
A fine topology has also been constructed in axiomatic potential theory (cf. [[#References|[3]]]).
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Brélot,  "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris  (1959)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.S. Landkof,  "Foundations of modern potential theory" , Springer  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Brélot,  "Lectures on potential theory" , Tata Inst.  (1960)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Brélot,  "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris  (1959)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.S. Landkof,  "Foundations of modern potential theory" , Springer  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Brélot,  "Lectures on potential theory" , Tata Inst.  (1960)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 10:15, 30 January 2022


in potential theory

The weakest topology in which all superharmonic functions on $ \mathbf R ^ {n} $ are continuous. Objects related to the fine topology are described as "fine" , "finely" , etc.

The notion of fine topology is closely connected with that of a thin set (cf. also Thinness of a set). The fine topology is stronger that the usual Euclidean topology on $ \mathbf R ^ {n} $, i.e. every Euclidean-open set is finely open. A fine neighbourhood of a point $ x _ {0} \in \mathbf R ^ {n} $ is a set $ V ( x _ {0} ) $ such that $ x _ {0} \in V ( x _ {0} ) $ and such that the complement $ {C V ( x _ {0} ) } $ is a thin set at $ x $. The finely-open sets are unions of pre-images under mappings by superharmonic functions of the extended real line $ \overline{\mathbf R} $ and of intervals of the form $ ( a, + \infty ] $, $ [- \infty , b) $, $ ( a, b) $, $ - \infty < a < b < + \infty $. Every superharmonic function on an open set $ E \subset \mathbf R ^ {n} $ is finely continuous on $ E $. A set $ E \subset \mathbf R ^ {n} $ is thin at a point $ x _ {0} \in E $ if and only if $ x _ {0} $ is a finely-isolated point of $ E $.

Let $ x _ {0} $ be a fine-limit point of $ E $, that is, $ E $ is not thin at $ x _ {0} $, and let $ f $ be a function defined on $ E $. The number $ \lambda $ is called the fine limit of $ f $ at $ x _ {0} $ if for every neighbourhood $ U ( \lambda ) $ of $ \lambda $ in $ \overline{\mathbf R} $ there exists a fine neighbourhood $ V ( x _ {0} ) $ of $ x _ {0} $ such that

$$ x \in E \cap V ( x _ {0} ) \Rightarrow f ( x) \in U ( \lambda ). $$

If $ \lambda $ is the fine limit of $ f $ at $ x _ {0} $, then there exists a fine neighbourhood $ V ( x _ {0} ) $ such that $ \lambda $ is an ordinary limit at $ x _ {0} $ of the restriction $ f \mid _ {E \cap V ( x _ {0} ) } $ (Cartan's theorem).

Let $ E $ be a closed set, thin at a point $ x _ {0} $, and let $ f > 0 $ be a superharmonic function defined on $ C E $ in a neighbourhood of $ x _ {0} $. Then $ f $ has a fine limit $ \lambda $ at $ x _ {0} $.

A fine topology has also been constructed in axiomatic potential theory (cf. [3]).

References

[1] M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1959)
[2] N.S. Landkof, "Foundations of modern potential theory" , Springer (1972) (Translated from Russian)
[3] M. Brélot, "Lectures on potential theory" , Tata Inst. (1960)

Comments

A potential theory for finely-harmonic and finely-hyperharmonic functions is developed in [a1]. See also [a2].

References

[a1] B. Fuglede, "Finely harmonic functions" , Springer (1972)
[a2] J. Lukeš, J. Malý, L. Zajíček, "Fine topology methods in real analysis and potential theory" , Springer (1986)
How to Cite This Entry:
Fine topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fine_topology&oldid=19006
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article