Fine topology (potential theory)

In mathematics, in the field of potential theory, the fine topology is a natural topology for setting the study of subharmonic functions. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Potential theory may be defined as the study of Harmonic functions Definition and comments The term "potential theory" arises from the fact that In Mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in Partial differential equations In the earliest studies of subharmonic functions, only smooth functions were considered, namely those for which $\Delta u \ge 0,$ where $Δ$ is the Laplacian. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability In Mathematics and Physics, the Laplace operator or Laplacian, denoted by \Delta\ or \nabla^2 and named after In that case it was natural to consider only the Euclidean topology, but with the advent of upper semi-continuous subharmonic functions introduced by F. Riesz, the fine topology became the more natural tool in many situations. For the notion of upper or lower semicontinuous Multivalued function see Hemicontinuity In Mathematical analysis, semi-continuity Frigyes Riesz ( January 22, 1880 &ndash February 28, 1956) was a Mathematician who was born in Győr, Hungary

1 Definition
2 Observations
3 Properties of the fine topology
4 References

Definition

The fine topology on the Euclidean space $\R^n$ is defined to be the coarsest topology making all subharmonic functions (equivalently all superharmonic functions) continuous. In Topology and related areas of Mathematics comparison of topologies refers to the fact that two Topological structures on a given set may stand in relation Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In Mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in Partial differential equations In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output Concepts in the fine topology are normally prefixed with the word 'fine' to distinguish them from the corresponding concepts in the usual topology, as for example 'fine neighbourhood' or 'fine continuous'.

Observations

The fine topology was introduced in 1940 by Henri Cartan to aid in the study of thin sets and was initially considered to be somewhat pathological due to the absence of a number of properties such as local compactness which are so frequently useful in analysis. Henri Paul Cartan ( July 8, 1904 &ndash August 13, 2008) was a son of Élie Cartan, and was as his father was a distinguished Subsequent work has shown that the lack of such properties is to a certain extent compensated for by the presence of other slightly less strong properties such as the quasi-Lindelöf property.

In one dimension, that is, on the real line, the fine topology coincides with the usual topology since in that case the subharmonic functions are precisely the convex functions which are already continuous in the usual (Euclidean) topology. In Mathematics, the real line is simply the set R of singleton Real numbers However this term is usually used when R is to be treated as a In Mathematics, a real-valued function f defined on an interval (or on any Convex subset of some Vector space) is called convex Thus, the fine topology is of most interest in $\R^n$ where $n\geq 2$ . The fine topology in this case is strictly finer than the usual topology, since there are discontinuous subharmonic functions.

Cartan observed in correspondence with Marcel Brelot that it is equally possible to develop the theory of the fine topology by using the concept of 'thinness'. In this development, a set $U$ is thin at a point $ζ$ if there exists a subharmonic function $v$ defined on a neighbourhood of $ζ$ such that

$v(\zeta)>\limsup_{z\to\zeta, z\in U} v(z).$

Then, a set $U$ is a fine neighbourhood of $ζ$ if and only if the complement of $U$ is thin at $ζ$ .

Properties of the fine topology

The fine topology is in some ways much less tractable than the usual topology in euclidean space, as is evidenced by the following (taking $n \ge 2$ ):

A set $F$ in $\R^n$ is fine compact if and only if $F$ is finite.
The fine topology on $\R^n$ is not locally compact (although it is Hausdorff). In Topology and related branches of Mathematics, a Topological space is called locally compact if roughly speaking each small portion of the space looks In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space
The fine topology on $\R^n$ is not first-countable, second-countable or metrisable. In Topology, a branch of Mathematics, a first-countable space is a Topological space satisfying the "first Axiom of countability " In Topology, a second-countable space is a Topological space satisfying the " second Axiom of countability " In Topology and related areas of Mathematics, a metrizable space is a Topological space that is homeomorphic to a Metric space.

The fine topology does at least have a few 'nicer' properties:

The fine topology has the Baire property. A subset A of a Topological space X has the property of Baire ( Baire property) if it differs from an Open set by a Meager
The fine topology in $\R^n$ is locally connected. In Topology and other branches of Mathematics, a Topological space is said to be locally connected at x (where x is a point

The fine topology does not possess the Lindelöf property but it does have the slightly weaker quasi-Lindelöf property:

An arbitrary union of fine open subsets of $\R^n$ differs by a polar set from some countable subunion. In Mathematics, a Lindelöf space is a Topological space in which every Open cover has a countable subcover In Mathematics, in the area of classical Potential theory, polar sets are the "negligible sets" similar to the way in which sets of measure zero are the

References

J. L. Doob. Classical Potential Theory and Its Probabilistic Counterpart, Springer-Verlag, Berlin Heidelberg New York, ISBN 3-540-41206-9.
L. L. Helms (1975). Introduction to potential theory. R. E. Krieger ISBN 0-88275-224-3.

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Contents

Definition

Observations

Properties of the fine topology

References