Separable space

Last updated
Separation axioms
in topological spaces
Kolmogorov classification
T0  (Kolmogorov)
T1  (Fréchet)
T2  (Hausdorff)
T2½ (Urysohn)
completely T2  (completely Hausdorff)
T3  (regular Hausdorff)
T3½ (Tychonoff)
T4  (normal Hausdorff)
T5  (completely normal
Hausdorff)
T6  (perfectly normal
Hausdorff)

In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence ${\displaystyle \{x_{n}\}_{n=1}^{\infty }}$ of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.

Contents

Like the other axioms of countability, separability is a "limitation on size", not necessarily in terms of cardinality (though, in the presence of the Hausdorff axiom, this does turn out to be the case; see below) but in a more subtle topological sense. In particular, every continuous function on a separable space whose image is a subset of a Hausdorff space is determined by its values on the countable dense subset.

Contrast separability with the related notion of second countability, which is in general stronger but equivalent on the class of metrizable spaces.

First examples

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. Similarly the set of all vectors ${\displaystyle {\boldsymbol {r}}=(r_{1},\ldots ,r_{n})\in \mathbb {R} ^{n}}$ of which ${\displaystyle {\boldsymbol {r}}\in \mathbb {Q} ^{n}}$ is a countable dense subset; so for every ${\displaystyle n}$, ${\displaystyle n}$-dimensional Euclidean space is separable.

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 ${\displaystyle \{U_{n}\}}$ is a countable base, choosing any ${\displaystyle x_{n}\in U_{n}}$ from the non-empty ${\displaystyle U_{n}}$ 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 subspace of 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 ( Willard 1970 , Th. 16.4a); even a quotient of 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 ${\displaystyle X}$, pick some ${\displaystyle x_{0}\in X}$, and define the topology to be the collection of all sets that contain ${\displaystyle x_{0}}$ (or are empty). Then, the closure of ${\displaystyle {x_{0}}}$ is the whole space (${\displaystyle X}$ is the smallest closed set containing ${\displaystyle x_{0}}$), but every set of the form ${\displaystyle \{x_{0},x\}}$ is open. Therefore, the space is separable but there cannot be a countable base.

Cardinality

The property of separability does not in and of itself give any limitations on the cardinality of a topological space: any set endowed with the trivial topology is separable, as well as second countable, quasi-compact, and connected. The "trouble" with the trivial topology is its poor separation properties: its Kolmogorov quotient is the one-point space.

A first-countable, separable Hausdorff space (in particular, a separable metric space) has at most the continuum cardinality ${\displaystyle {\mathfrak {c}}}$. In such a space, closure is determined by limits of sequences and any convergent sequence has at most one limit, so there is a surjective map from the set of convergent sequences with values in the countable dense subset to the points of ${\displaystyle X}$.

A separable Hausdorff space has cardinality at most ${\displaystyle 2^{\mathfrak {c}}}$, where ${\displaystyle {\mathfrak {c}}}$ is the cardinality of the continuum. For this closure is characterized in terms of limits of filter bases: if ${\displaystyle Y\subseteq X}$ and ${\displaystyle z\in X}$, then ${\displaystyle z\in {\overline {Y}}}$ if and only if there exists a filter base ${\displaystyle {\mathcal {B}}}$ consisting of subsets of ${\displaystyle Y}$ that converges to ${\displaystyle z}$. The cardinality of the set ${\displaystyle S(Y)}$ of such filter bases is at most ${\displaystyle 2^{2^{|Y|}}}$. Moreover, in a Hausdorff space, there is at most one limit to every filter base. Therefore, there is a surjection ${\displaystyle S(Y)\rightarrow X}$ when ${\displaystyle {\overline {Y}}=X.}$

The same arguments establish a more general result: suppose that a Hausdorff topological space ${\displaystyle X}$ contains a dense subset of cardinality ${\displaystyle \kappa }$. Then ${\displaystyle X}$ has cardinality at most ${\displaystyle 2^{2^{\kappa }}}$ and cardinality at most ${\displaystyle 2^{\kappa }}$ 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 ${\displaystyle \mathbb {R} ^{\mathbb {R} }}$ of all functions from the real line to itself, endowed with the product topology, is a separable Hausdorff space of cardinality ${\displaystyle 2^{\mathfrak {c}}}$. More generally, if ${\displaystyle \kappa }$ is any infinite cardinal, then a product of at most ${\displaystyle 2^{\kappa }}$ spaces with dense subsets of size at most ${\displaystyle \kappa }$ has itself a dense subset of size at most ${\displaystyle \kappa }$ (Hewitt–Marczewski–Pondiczery theorem).

Constructive mathematics

Separability is especially important in numerical analysis and constructive mathematics, since many theorems that can be proved for nonseparable spaces have constructive proofs only for separable spaces. Such constructive proofs can be turned into algorithms for use in numerical analysis, and they are the only sorts of proofs acceptable in constructive analysis. A famous example of a theorem of this sort is the Hahn–Banach theorem.

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 ${\displaystyle n}$-dimensional Euclidean space is separable.
• The space ${\displaystyle C(K)}$ of all continuous functions from a compact subset ${\displaystyle K\subseteq \mathbb {R} }$ to the real line ${\displaystyle \mathbb {R} }$ is separable.
• The Lebesgue spaces ${\displaystyle L^{p}\left(X,\mu \right)}$, over a separable measure space ${\displaystyle \left\langle X,{\mathcal {M}},\mu \right\rangle }$, are separable for any ${\displaystyle 1\leq p<\infty }$.
• The space ${\displaystyle C([0,1])}$ of continuous real-valued functions on the unit interval ${\displaystyle [0,1]}$ with the metric of uniform convergence is a separable space, since it follows from the Weierstrass approximation theorem that the set ${\displaystyle \mathbb {Q} [x]}$ of polynomials in one variable with rational coefficients is a countable dense subset of ${\displaystyle C([0,1])}$. The Banach–Mazur theorem asserts that any separable Banach space is isometrically isomorphic to a closed linear subspace of ${\displaystyle C([0,1])}$.
• 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 ${\displaystyle \ell ^{2}}$ of square-summable sequences.
• An example of a separable space that is not second-countable is the Sorgenfrey line ${\displaystyle \mathbb {S} }$, the set of real numbers equipped with the lower limit topology.
• A separable σ-algebra is a σ-algebra ${\displaystyle {\mathcal {F}}}$ that is a separable space when considered as a metric space with metric ${\displaystyle \rho (A,B)=\mu (A\triangle B)}$ for ${\displaystyle A,B\in {\mathcal {F}}}$ and a given measure ${\displaystyle \mu }$ (and with ${\displaystyle \triangle }$ being the symmetric difference operator). [1]

Non-separable spaces

• The first uncountable ordinal ${\displaystyle \omega _{1}}$, equipped with its natural order topology, is not separable.
• The Banach space ${\displaystyle \ell ^{\infty }}$ of all bounded real sequences, with the supremum norm, is not separable. The same holds for ${\displaystyle L^{\infty }}$.
• 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 space is 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 less than or equal to ${\displaystyle {\mathfrak {c}}}$. 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:
(i) X is second countable.
(ii) The space ${\displaystyle {\mathcal {C}}(X,\mathbb {R} )}$ of continuous real-valued functions on X with the supremum norm is separable.
(iii) X is metrizable.

Embedding separable metric spaces

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 1969 )

Related Research Articles

In mathematics, more specifically in functional analysis, a Banach space is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space.

In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M.

In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space is said to be metrizable if there is a metric such that the topology induced by is . Metrization theorems are theorems that give sufficient conditions for a topological space to be metrizable.

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.

In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. A topological vector space is a vector space which is also a topological space, this implies that vector space operations be continuous functions. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence.

The Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space.

In mathematics, a Baire space is a topological space such that every intersection of a countable collection of open dense sets in the space is also dense. Complete metric spaces and locally compact Hausdorff spaces are examples of Baire spaces according to the Baire category theorem. The spaces are named in honor of René-Louis Baire who introduced the concept.

In the mathematical fields of general topology and descriptive set theory, a meagre set is a set that, considered as a subset of a topological space, is in a precise sense small or negligible. A topological space T is called meagre if it is a meager subset of itself; otherwise, it is called nonmeagre.

In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces. All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically not Banach spaces.

In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of compactness, which requires the existence of a finite subcover.

In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set. In set theory, the topological space 2ω is called "the" Cantor space.

In the mathematical discipline of general 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 dense subset. Polish spaces are so named because they were first extensively studied by Polish topologists and logicians—Sierpiński, Kuratowski, Tarski and others. However, Polish spaces are mostly studied today because they are the primary setting for descriptive set theory, including the study of Borel equivalence relations. Polish spaces are also a convenient setting for more advanced measure theory, in particular in probability theory.

In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.

In functional analysis and related branches of mathematics, the Banach–Alaoglu theorem states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common proof identifies the unit ball with the weak-* topology as a closed subset of a product of compact sets with the product topology. As a consequence of Tychonoff's theorem, this product, and hence the unit ball within, is compact.

In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space is second-countable if there exists some countable collection of open subsets of such that any open subset of can be written as a union of elements of some subfamily of . A second-countable space is said to satisfy the second axiom of countability. Like other countability axioms, the property of being second-countable restricts the number of open sets that a space can have.

In mathematics, a regular measure on a topological space is a measure for which every measurable set can be approximated from above by open measurable sets and from below by compact measurable sets.

In mathematics, the Moore plane, also sometimes called Niemytzki plane, is a topological space. It is a completely regular Hausdorff space that is not normal. It is named after Robert Lee Moore and Viktor Vladimirovich Nemytskii.

In topology and related areas of mathematics, a subset A of a topological space X is called dense if every point x in X either belongs to A or is a limit point of A; that is, the closure of A constitutes the whole set X. Informally, for every point in X, the point is either in A or arbitrarily "close" to a member of A — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it.

In mathematics, in the areas of topology and functional analysis, the Anderson–Kadec theorem states that any two infinite-dimensional, separable Banach spaces, or, more generally, Fréchet spaces, are homeomorphic as topological spaces. The theorem was proved by Mikhail Kadets (1966) and Richard Davis Anderson.

In functional analysis and related areas of mathematics, a metrizable topological vector spaces (TVS) is a TVS whose topology is induced by a metric. An LM-space is an inductive limit of a sequence of locally convex metrizable TVS.

References

1. Džamonja, Mirna; Kunen, Kenneth (1995). "Properties of the class of measure separable compact spaces" (PDF). Fundamenta Mathematicae : 262. arXiv:. Bibcode:1994math......8201D. If ${\displaystyle \mu }$ is a Borel measure on ${\displaystyle X}$, the measure algebra of ${\displaystyle (X,\mu )}$ is the Boolean algebra of all Borel sets modulo ${\displaystyle \mu }$-null sets. If ${\displaystyle \mu }$ 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 ${\displaystyle \mu }$ is separable iff this metric space is separable as a topological space.