Fine Topology (other)
In mathematics, fine topology can refer to: * Fine topology (potential theory) * The sense opposite to coarse topology, namely: ** A term in comparison of topologies which specifies the partial order relation of a topological structure to other one(s) ** Final topology See also * Discrete topology In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest t ...

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. In the earliest studies of subharmonic functions, namely those for which \Delta u \ge 0, where \Delta is the Laplacian, only smooth functions were considered. 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. 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. 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 ...
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Coarse Topology (other)
In mathematics, coarse topology is a term in comparison of topologies which specifies the partial order relation of a topological structure to other one(s). Specifically, the coarsest topology may refer to: * Initial topology, the most coarse topology in a certain category of topologies * Trivial topology, the most coarse topology possible on a given set See also * Weak topology, an example of topology coarser than the standard one * Fine topology (other) In mathematics, fine topology can refer to: * Fine topology (potential theory) * The sense opposite to coarse topology, namely: ** A term in comparison of topologies which specifies the partial order relation of a topological structure to other on ...

Comparison Of Topologies
In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies. Definition A topology on a set may be defined as the collection of subsets which are considered to be "open". An alternative definition is that it is the collection of subsets which are considered "closed". These two ways of defining the topology are essentially equivalent because the complement of an open set is closed and vice versa. In the following, it doesn't matter which definition is used. Let ''τ''1 and ''τ''2 be two topologies on a set ''X'' such that ''τ''1 is contained in ''τ''2: :\tau_1 \subseteq \tau_2. That is, every element of ''τ''1 is also an element of ''τ''2. Then the topology ''τ''1 is said to be a coarser (weaker or smaller) topology than ''τ''2, and ''τ''2 is said to be a finer (stronger or larger) topology than ''τ''1. There are some authors, especially ...
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Final Topology
In general topology and related areas of mathematics, the final topology (or coinduced, strong, colimit, or inductive topology) on a set X, with respect to a family of functions from topological spaces into X, is the finest topology on X that makes all those functions continuous. The quotient topology on a quotient space is a final topology, with respect to a single surjective function, namely the quotient map. The disjoint union topology is the final topology with respect to the inclusion maps. The final topology is also the topology that every direct limit in the category of topological spaces is endowed with, and it is in the context of direct limits that the final topology often appears. A topology is coherent with some collection of subspaces if and only if it is the final topology induced by the natural inclusions. The dual notion is the initial topology, which for a given family of functions from a set X into topological spaces is the coarsest topology on X ...
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