# Fine topology

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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)