|
|
| Line 1: |
Line 1: |
| In [[mathematics]], in the field of [[potential theory]], the '''fine topology''' is a [[natural topology]] for setting the study of [[subharmonic function]]s. In the earliest studies of subharmonic functions, namely those for which <math>\Delta u \ge 0,</math> where <math>\Delta</math> is the [[Laplacian]], only [[smooth function]]s were considered. In that case it was natural to consider only the [[Euclidean space|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.
| | Her title is Felicidad Ahmad. My home is now in Kansas. One of his preferred hobbies is taking part in crochet but he hasn't produced a dime with it. My occupation is a production and distribution officer and I'm doing fairly good financially.<br><br>Review my blog [http://www.alexsantosdesign.com/index.php?mod=users&action=view&id=8657 extended auto warranty] |
| | |
| == Definition ==
| |
| | |
| The fine topology on the [[Euclidean space]] <math>\R^n</math> is defined to be the
| |
| [[comparison of topologies|coarsest]] [[topology]] making all [[subharmonic function]]s (equivalently all superharmonic functions) [[continuous function|continuous]]. 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 set (analysis)|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. 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 function]]s which are already continuous in the usual (Euclidean) topology. Thus, the fine topology is of most interest in <math>\R^n</math> where <math>n\geq 2</math>. 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 <math>U</math> is '''thin''' at a point <math>\zeta</math> if there exists a subharmonic function <math>v</math> defined on a neighbourhood of <math>\zeta</math> such that
| |
| | |
| :<math>v(\zeta)>\limsup_{z\to\zeta, z\in U} v(z).</math>
| |
| | |
| Then, a set <math>U</math> is a fine neighbourhood of <math>\zeta</math> if and only if the complement of <math>U</math> is thin at <math>\zeta</math>.
| |
| | |
| == 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 <math>n \ge 2</math>):
| |
| | |
| *A set <math>F</math> in <math>\R^n</math> is fine [[compact set|compact]] if and only if <math>F</math> is finite.
| |
| *The fine topology on <math>\R^n</math> is not [[locally compact]] (although it is [[Hausdorff space|Hausdorff]]).
| |
| *The fine topology on <math>\R^n</math> is not [[first-countable space|first-countable]], [[second-countable space|second-countable]] or [[metrisable]].
| |
| | |
| The fine topology does at least have a few 'nicer' properties:
| |
| | |
| *The fine topology has the [[Baire property]].
| |
| *The fine topology in <math>\R^n</math> is [[locally connected]].
| |
| | |
| The fine topology does not possess the [[Lindelöf space|Lindelöf property]] but it does have the slightly weaker quasi-Lindelöf property:
| |
| | |
| *An arbitrary union of fine open subsets of <math>\R^n</math> differs by a [[Polar set (potential theory)|polar set]] from some countable subunion.
| |
| | |
| == References ==
| |
| *{{citation | first=John B. | last=Conway | authorlink=John B. Conway | isbn=0-387-94460-5 | series=[[Graduate Texts in Mathematics]] | volume=159 | title=Functions of One Complex Variable II | publisher=[[Springer-Verlag]] | pages=367–376 }}
| |
| * {{citation | first=J. L. | last=Doob | authorlink=Joseph Leo Doob | title=Classical Potential Theory and Its Probabilistic Counterpart | publisher=Springer-Verlag | location=Berlin Heidelberg New York | isbn=3-540-41206-9 }}
| |
| *{{citation | first=L. L. | last=Helms | year=1975 | title=Introduction to potential theory | publisher=R. E. Krieger | isbn=0-88275-224-3 }}
| |
| | |
| [[Category:Subharmonic functions]]
| |
Her title is Felicidad Ahmad. My home is now in Kansas. One of his preferred hobbies is taking part in crochet but he hasn't produced a dime with it. My occupation is a production and distribution officer and I'm doing fairly good financially.
Review my blog extended auto warranty