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Splunk Inc. is an American software company based in San Francisco, California, [2] that produces software for searching, monitoring, ...
The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. This is a list of topology topics. See also: Topology glossary; List of topologies; List of general topology topics; List of geometric topology topics
The finest topology on X is the discrete topology; this topology makes all subsets open. The coarsest topology on X is the trivial topology; this topology only admits the empty set and the whole space as open sets. In function spaces and spaces of measures there are often a number of possible topologies.
In mathematics, pointless topology, also called point-free topology (or pointfree topology) and locale theory, is an approach to topology that avoids mentioning points, and in which the lattices of open sets are the primitive notions. [1] In this approach it becomes possible to construct topologically interesting spaces from purely algebraic ...
The product topology on is equal to the initial topology induced by the canonical projections : as ranges over . [4] Consequently, the initial topology on induced by {:} is equal to the inverse image of the product topology on by the evaluation map:. [4] Furthermore, if the maps {} separate points on then the evaluation map is a homeomorphism ...
The following sets will constitute the basic open subsets of topologies on spaces of linear maps. For any subsets and , let (,):= {: ()}.. The family {(,):,} forms a neighborhood basis [1] at the origin for a unique translation-invariant topology on , where this topology is not necessarily a vector topology (that is, it might not make into a TVS).
The archetypical example of a filter is the neighborhood filter at a point in a topological space (,), which is the family of sets consisting of all neighborhoods of . By definition, a neighborhood of some given point is any subset whose topological interior contains this point; that is, such that . Importantly, neighborhoods are not required to be open sets; those are called open ...
Lately, topology control algorithms have been divided into two subproblems: topology construction, in charge of the initial reduction, and topology maintenance, in charge of the maintenance of the reduced topology so that characteristics like connectivity and coverage are preserved. This is the first stage of a topology control protocol.