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. 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. Thus, the fine topology is of most interest in where . 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 is thin at a point if there exists a subharmonic function defined on a neighbourhood of such that
Then, a set is a fine neighbourhood of if and only if the complement of is thin at .
Read more about this topic: Fine Topology (potential Theory)
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