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Any topological space that is itself finite or countably infinite is separable, for the whole space is a countable dense subset of itself. An important example of an uncountable separable space is the real line, in which the rational numbers form a countable dense subset.
Separability may refer to: Mathematics. Separable algebra, a generalization to associative algebras of the notion of a separable field extension;
In Euclidean geometry, linear separability is a property of two sets of points. This is most easily visualized in two dimensions (the Euclidean plane ) by thinking of one set of points as being colored blue and the other set of points as being colored red.
The separability problem is a subject of current research. A separability criterion is a necessary condition a state must satisfy to be separable. In the low-dimensional ( 2 X 2 and 2 X 3 ) cases, the Peres-Horodecki criterion is actually a necessary and sufficient condition for separability.
Separability problems may arise when dealing with transcendental extensions. This is typically the case for algebraic geometry over a field of prime characteristic, where the function field of an algebraic variety has a transcendence degree over the ground field that is equal to the dimension of the variety.
The separability condition above will imply every finitely generated A-module M is isomorphic to a direct summand in its restricted, induced module. But if B has finite representation type, the restricted module is uniquely a direct sum of multiples of finitely many indecomposables , which induce to a finite number of constituent indecomposable ...
In the multipartite case there is no simple necessary and sufficient condition for separability like the one given by the PPT criterion for the and cases. However, many separability criteria used in the bipartite setting can be generalized to the multipartite case.
In the 2×2 and 2×3 dimensional cases the condition is also sufficient. It is used to decide the separability of mixed states, where the Schmidt decomposition does not apply. The theorem was discovered in 1996 by Asher Peres [1] and the Horodecki family (Michał, Paweł, and Ryszard) [2]