Separable space
In
Like the other
Contrast separability with the related notion of
First examples
Any topological space that is itself
A simple example of a space that is not separable is a discrete space of uncountable cardinality.
Further examples are given below.
Separability versus second countability
Any second-countable space is separable: if is a countable base, choosing any from the non-empty gives a countable dense subset. Conversely, a metrizable space is separable if and only if it is second countable, which is the case if and only if it is Lindelöf.
To further compare these two properties:
- An arbitrary subspaceof a second-countable space is second countable; subspaces of separable spaces need not be separable (see below).
- Any continuous image of a separable space is separable (quotientof a second-countable space need not be second countable.
- A product of at most continuum many separable spaces is separable (Willard 1970, p. 109, Th 16.4c). A countable product of second-countable spaces is second countable, but an uncountable product of second-countable spaces need not even be first countable.
We can construct an example of a separable topological space that is not second countable. Consider any uncountable set , pick some , and define the topology to be the collection of all sets that contain (or are empty). Then, the closure of is the whole space ( is the smallest closed set containing ), but every set of the form is open. Therefore, the space is separable but there cannot have a countable base.
Cardinality
The property of separability does not in and of itself give any limitations on the
A
A separable Hausdorff space has cardinality at most , where is the cardinality of the continuum. For this closure is characterized in terms of limits of filter bases: if and , then if and only if there exists a filter base consisting of subsets of that converges to . The cardinality of the set of such filter bases is at most . Moreover, in a Hausdorff space, there is at most one limit to every filter base. Therefore, there is a surjection when
The same arguments establish a more general result: suppose that a Hausdorff topological space contains a dense subset of cardinality . Then has cardinality at most and cardinality at most if it is first countable.
The product of at most continuum many separable spaces is a separable space (Willard 1970, p. 109, Th 16.4c). In particular the space of all functions from the real line to itself, endowed with the product topology, is a separable Hausdorff space of cardinality . More generally, if is any infinite cardinal, then a product of at most spaces with dense subsets of size at most has itself a dense subset of size at most (Hewitt–Marczewski–Pondiczery theorem).
Constructive mathematics
Separability is especially important in
Further examples
Separable spaces
- Every compact metric space (or metrizable space) is separable.
- Any topological space that is the union of a countable number of separable subspaces is separable. Together, these first two examples give a different proof that -dimensional Euclidean space is separable.
- The space of all continuous functions from a compact subset to the real line is separable.
- The Lebesgue spaces , over a measure space whose σ-algebra is countably generated and whose measure is σ-finite, are separable for any .[1]
- The space of continuous real-valued functions on the unit interval with the metric of uniform convergence is a separable space, since it follows from the Weierstrass approximation theorem that the set of polynomials in one variable with rational coefficients is a countable dense subset of . The Banach–Mazur theorem asserts that any separable Banach space is isometrically isomorphic to a closed linear subspace of .
- A Hilbert space is separable if and only if it has a countable orthonormal basis. It follows that any separable, infinite-dimensional Hilbert space is isometric to the space of square-summable sequences.
- An example of a separable space that is not second-countable is the Sorgenfrey line, the set of real numbers equipped with the lower limit topology.
- A separable σ-algebra is a σ-algebra that is a separable space when considered as a metricfor and a given finite measure (and with being the symmetric difference operator).[2]
Non-separable spaces
- The first uncountable ordinal , equipped with its natural order topology, is not separable.
- The Banach space of all bounded real sequences, with the supremum norm, is not separable. The same holds for .
- The Banach space of functions of bounded variation is not separable; note however that this space has very important applications in mathematics, physics and engineering.
Properties
- A subspace of a separable space need not be separable (see the Sorgenfrey plane and the Moore plane), but every open subspace of a separable space is separable (Willard 1970, Th 16.4b). Also every subspace of a separable metric spaceis separable.
- In fact, every topological space is a subspace of a separable space of the same cardinality. A construction adding at most countably many points is given in (Sierpiński 1952, p. 49); if the space was a Hausdorff space then the space constructed that it embeds into is also a Hausdorff space.
- The set of all real-valued continuous functions on a separable space has a cardinality equal to , the cardinality of the continuum. This follows since such functions are determined by their values on dense subsets.
- From the above property, one can deduce the following: If X is a separable space having an uncountable closed discrete subspace, then X cannot be normal. This shows that the Sorgenfrey plane is not normal.
- For a compact Hausdorff space X, the following are equivalent:
- X is second countable.
- The space of continuous real-valued functions on X with the supremum norm is separable.
- X is metrizable.
Embedding separable metric spaces
- Every separable metric space is Urysohn metrization theorem.
- Every separable metric space is isometric to a subset of the (non-separable) Banach space l∞ of all bounded real sequences with the supremum norm; this is known as the Fréchet embedding. (Heinonen 2003)
- Every separable metric space is isometric to a subset of C([0,1]), the separable Banach space of continuous functions [0,1] → R, with the supremum norm. This is due to Stefan Banach. (Heinonen 2003)
- Every separable metric space is isometric to a subset of the Urysohn universal space.
For nonseparable spaces:
- A metric space of density equal to an infinite cardinal α is isometric to a subspace of C([0,1]α, R), the space of real continuous functions on the product of α copies of the unit interval. (Kleiber & Pervin 1969)
References
- ^ Donald L. Cohn (2013). Measure Theory. Springer Science+Business Media., Proposition 3.4.5.
- Bibcode:1994math......8201D.
If is a Borel measure on , the measure algebra of is the Boolean algebra of all Borel sets modulo -null sets. If is finite, then such a measure algebra is also a metric space, with the distance between the two sets being the measure of their symmetric difference. Then, we say that is separable iff this metric space is separable as a topological space.
- Heinonen, Juha (January 2003), Geometric embeddings of metric spaces (PDF), retrieved 6 February 2009
- MR 0370454
- Kleiber, Martin; Pervin, William J. (1969), "A generalized Banach-Mazur theorem", Bull. Austral. Math. Soc., 1 (2): 169–173,
- MR 0050870
- MR 0507446
- Willard, Stephen (1970), General Topology, MR 0264581