In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence ( x n ) n = 1 ∞ {\displaystyle...

of space. From this perspective, the natural state space of a boson might seem to be a non-separable space. However, it is only a small separable subspace...

roots is equal to its degree Separable sigma algebra, a separable space in measure theory Separable space, a topological space that contains a countable...

second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T...

topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable...

of a separable Banach space need not be separable, but: Theorem—Let X {\displaystyle X} be a normed space. If X ′ {\displaystyle X'} is separable, then...

higher than continuum). A separable measure space has a natural pseudometric that renders it separable as a pseudometric space. The distance between two...

Hausdorff space is metrizable if and only if it is second-countable. Urysohn's Theorem can be restated as: A topological space is separable and metrizable...

Fréchet–Urysohn space – Property of topological space Second-countable space – Topological space whose topology has a countable base Separable space – Topological...

infinite-dimensional spaces due to a key limitation: any translation-invariant Borel measure on an infinite-dimensional separable Banach space must be either...

space is Polish if it is separable and completely metrizable, i.e. if it is homeomorphic to a separable and complete metric space. Polyadic A space is...

subset of this space, and this is a non-separable subset of the separable space S {\displaystyle \mathbb {S} } . It shows that separability does not inherit...

In quantum mechanics, separable states are multipartite quantum states that can be written as a convex combination of product states. Product states are...

p}(\Omega )} is a Banach space. For p < ∞ , W k , p ( Ω ) {\displaystyle p<\infty ,W^{k,p}(\Omega )} is also a separable space. It is conventional to denote...

{\hat {A}}\cong \operatorname {Prim} (A).} Let H be a separable infinite-dimensional Hilbert space. L(H) has two norm-closed *-ideals: I0 = {0} and the...

space is a function whose composition with any element of the dual space is a measurable function in the usual (strong) sense. For separable spaces,...

on separable spaces and most applications to other areas of mathematics or physics only use separable Hilbert spaces. Note that if the measure space (X...

a separable extension if for every α ∈ E {\displaystyle \alpha \in E} , the minimal polynomial of α {\displaystyle \alpha } over F is a separable polynomial...

For a stochastic process to be separable, in addition to other conditions, its index set must be a separable space, which means that the index set has...

construction of Tsirelson space in 1974. The dual statement, that every separable Banach space is linearly isometric to a quotient space of ℓ1, was answered...

Arzelà–Ascoli theorem. A metric space is separable if and only if it is homeomorphic to a totally bounded metric space. The closure of a totally bounded...

should not be confused with separated spaces (defined below), which are somewhat related but different. Separable spaces are again a completely different topological...

be found in (Schwartz 1969, p. 21). If H1 is a separable space (in particular, if it is a Euclidean space) the result is true in Zermelo–Fraenkel set theory;...

\sigma _{0}} , D {\displaystyle \mathbb {D} } is a separable space. Thus, Skorokhod space is a Polish space. By an application of the Arzelà–Ascoli theorem...

Marczewski proved that the topological dimension, for arbitrary metrisable separable space X, coincides with the Hausdorff dimension under one of the metrics...

notions above agree for separable, metrisable spaces.[citation needed][clarification needed] A zero-dimensional Hausdorff space is necessarily totally...

Problem. Every separable topological space has ccc. Furthermore, a product space of arbitrary amount of separable spaces has ccc. A metric space has ccc if...

requirement that R contains a countable dense subset (i.e., R is a separable space), then the answer is indeed yes: any such set R is necessarily order-isomorphic...

Any separable inner product space has an orthonormal basis. Using the Hausdorff maximal principle and the fact that in a complete inner product space orthogonal...

differential equation for the unknown f ( x ) {\displaystyle f(x)} is separable if it can be written in the form d d x f ( x ) = g ( x ) h ( f ( x ) )...