Separable space

In topology, a separable space is a topological space that possesses a countable dense subset, meaning there exists a countable collection of points whose closure is the entire space.^[1] This property, first introduced by Maurice Fréchet in 1906 in the context of functional analysis and metric spaces, captures the idea of a space that can be "separated" or approximated by a countable set of elements, such as the rational numbers in the real line. The term "separable" reflects this countable approximation, distinguishing such spaces from non-separable ones like the uncountable discrete space on an uncountable set.^[2] Separable spaces are fundamental in analysis and topology due to their connections with other key properties. For instance, every second-countable space— one with a countable basis for its topology—is separable, as selecting one point from each basis element yields a countable dense set.^[3] Conversely, in metric spaces, separability is equivalent to second-countability, ensuring that the topology can be generated by countably many open balls.^[4] Examples abound in familiar settings: the Euclidean space $\mathbb{R}^n$Rn with the standard topology is separable, with the rationals $\mathbb{Q}^n$Qn serving as a countable dense subset, and more generally, any Polish space—a separable completely metrizable space—is separable by definition.^[1][5] Notable aspects include inheritance properties and counterexamples that highlight limitations. While every open subspace of a separable space is separable, arbitrary subspaces need not be; for example, the Sorgenfrey plane (the product of two Sorgenfrey lines) is separable but contains non-separable subspaces.^[6] Separable metric spaces enjoy strong regularity: they are second-countable, hence paracompact and Lindelöf.^[4] They often embed into the Hilbert cube. These features make separability crucial for theorems in functional analysis, such as the Baire category theorem applied to complete separable spaces, and in geometric topology for studying manifolds and embeddings.^[5]

Definition

Formal definition

A topological space $X$X is separable if it contains a countable dense subset $D \subseteq X$D⊆X.^[1][4] This definition presupposes familiarity with the basic elements of topology, including the collection of open sets that form the topology on $X$X and the closure operator $\mathrm{cl}$cl, which assigns to each subset its smallest closed superset.^[6] A subset $D$D is dense in $X$X if $\mathrm{cl}(D) = X$cl(D)=X, meaning that $D$D intersects every nonempty open set $U \subseteq X$U⊆X (i.e., $U \cap D \neq \emptyset$U∩D=∅).^[4] Equivalently, the closure condition ensures that $D$D is "everywhere dense," with points of $D$D accumulating at every point of $X$X.^[6] Separability is inherently a topological property, determined solely by the open sets of the space and independent of the cardinality of the underlying set $X$X at the outset.^[1] For example, the rational numbers form a countable dense subset of the real numbers under the standard topology.^[4]

Equivalent characterizations

A topological space $X$X is separable if and only if its density character $\dens(X) \leq \aleph_0$\dens(X)≤ℵ0​, where the density character $\dens(X)$\dens(X) is defined as the smallest cardinality of any dense subset of $X$X. In sequential spaces, separability is equivalent to the existence of a countable dense subset $D \subseteq X$D⊆X such that every point of $X$X is the limit of a convergent sequence with terms in $D$D. This follows from the fact that in sequential spaces, the closure of a set coincides with its sequential closure.^[7] In any topological space, separability implies the existence of a countable subset $S \subseteq X$S⊆X that intersects every closed subset of $X$X with nonempty interior (take $S$S to be the countable dense subset itself, as it intersects the nonempty interior, which is open).

Examples

Basic separable spaces

Euclidean spaces provide a fundamental example of separable spaces. The space $\mathbb{R}^n$Rn, equipped with the standard Euclidean topology, is separable because the subset $\mathbb{Q}^n$Qn consisting of all points with rational coordinates in each component is countable and dense in $\mathbb{R}^n$Rn.^[8] This density follows from the fact that every open ball in $\mathbb{R}^n$Rn contains points with rational coordinates, allowing $\mathbb{Q}^n$Qn to approximate any point arbitrarily closely.^[9] Any countable topological space is inherently separable, as the space itself forms a countable dense subset.^[10] For instance, consider a countable set endowed with the discrete topology, where every subset is open; here, the entire space serves as the countable dense subset, since its closure is the space itself.^[11] In the realm of infinite-dimensional spaces, the Hilbert space $\ell^2$ℓ2 of square-summable real sequences is separable. A countable dense subset consists of all sequences with only finitely many nonzero entries, each of which is rational; this set is countable as a countable union over finite supports of countable products of rationals, and it is dense because any sequence in $\ell^2$ℓ2 can be approximated by truncating its tail and rationalizing its finite initial segment to within any desired norm.^[12]

Basic non-separable spaces

One basic example of a non-separable topological space is the uncountable discrete space, where an uncountable set is equipped with the discrete topology in which every subset is open.^[13] In this space, the singleton sets form an uncountable family of pairwise disjoint nonempty open sets. A countable dense subset would need to intersect every nonempty open set, but it can intersect at most countably many of these singletons, leaving uncountably many open singletons untouched by its closure, so no such dense subset exists.^[13] Another standard non-separable space is $\ell^\infty$ℓ∞, the Banach space of all bounded real-valued sequences equipped with the supremum norm $\|x\|_\infty = \sup_n |x_n|$∥x∥∞​=supn​∣xn​∣.^[14] To see its non-separability, consider the uncountable family of sequences that are the characteristic functions of singletons in an uncountable index set, or more simply, the set of all sequences with entries in $\{0,1\}${0,1}, which has cardinality $2^{\aleph_0}$2ℵ0​. The open balls of radius $1/2$1/2 around these distinct sequences are pairwise disjoint, as the supremum norm distance between any two distinct such sequences is 1. Thus, any countable subset can intersect at most countably many of these balls, implying no countable dense subset exists.^[14] The ordinal space $[0, \omega_1)$[0,ω1​) with the order topology provides a third fundamental example of non-separability. Here, $\omega_1$ω1​ is the first uncountable ordinal, and the topology is generated by open intervals $(\alpha, \beta)$(α,β) for ordinals $\alpha < \beta < \omega_1$α<β<ω1​. The singleton sets $\{\sigma\}${σ} for each successor ordinal $\sigma < \omega_1$σ<ω1​ (where $\sigma = \rho + 1$σ=ρ+1 for some $\rho < \omega_1$ρ<ω1​) form an uncountable collection of pairwise disjoint nonempty open sets, as each such singleton is the open interval $(\rho, \sigma + 1)$(ρ,σ+1). Since there are uncountably many successor ordinals below $\omega_1$ω1​, a countable dense subset cannot intersect all of them, preventing its closure from being the entire space. These examples illustrate the failure of separability through the presence of uncountably many pairwise disjoint nonempty open sets, which no countable set can densely approximate.

Relations to other separation axioms

Separability versus second countability

A second-countable space is a topological space that possesses a countable basis for its topology, meaning there exists a countable collection $\{B_n\}_{n \in \mathbb{N}}${Bn​}n∈N​ of open sets such that every open set in the space is a countable union of elements from this collection.^[15] Second countability implies separability in any topological space. To see this, consider the countable basis $\{B_n\}${Bn​}; by the axiom of countable choice, select a point $x_n \in B_n$xn​∈Bn​ for each $n$n. The set $D = \{x_n \mid n \in \mathbb{N}\}$D={xn​∣n∈N} is countable. For density, take any nonempty open set $U$U; then $U$U contains some basis element $B_k$Bk​, so $x_k \in U$xk​∈U, ensuring $D$D intersects every nonempty open set.^[16] However, separability does not imply second countability. A classic counterexample is the Sorgenfrey line, which is the real line $\mathbb{R}$R equipped with the lower limit topology generated by the basis of half-open intervals $\{[a, b) \mid a, b \in \mathbb{R}, a < b\}${[a,b)∣a,b∈R,a<b}.^[17] The Sorgenfrey line is separable because the set of rational numbers $\mathbb{Q}$Q is dense in it. For any nonempty basic open set $[a, b)$[a,b), the density of $\mathbb{Q}$Q in $\mathbb{R}$R with the standard topology guarantees a rational $q \in (a, b) \subseteq [a, b)$q∈(a,b)⊆[a,b), so $\mathbb{Q}$Q intersects every basic open set.^[18] Nevertheless, the Sorgenfrey line is not second countable. Suppose for contradiction that it has a countable basis $\mathcal{A} = \{A_n \mid n \in \mathbb{N}\}$A={An​∣n∈N}. For each $x \in \mathbb{R}$x∈R, the open interval $[x, x+1)$[x,x+1) contains $x$x and is itself a basic open set. Select $B_x \in \mathcal{A}$Bx​∈A such that $x \in B_x \subseteq [x, x+1)$x∈Bx​⊆[x,x+1). Any such $B_x$Bx​ must satisfy $\inf B_x = x$infBx​=x, because basis elements containing $x$x are of the form $[c, d)$[c,d) with $c \leq x < d$c≤x<d, and for $[c, d) \subseteq [x, x+1)$[c,d)⊆[x,x+1), it is necessary that $c = x$c=x. Thus, if $x \neq y$x=y, then $B_x \neq B_y$Bx​=By​. The map $x \mapsto B_x$x↦Bx​ is therefore an injection from the uncountable set $\mathbb{R}$R into the countable set $\mathcal{A}$A, a contradiction.^[19]

Separability in metric spaces

In metric spaces, separability is equivalent to second countability.^[10] This equivalence arises from the structure of the metric topology. If $(X, d)$(X,d) is a separable metric space, it contains a countable dense subset $\{x_n \mid n \in \mathbb{N}\}${xn​∣n∈N}. The collection of all open balls $B(x_n, q)$B(xn​,q) centered at these points with rational radii $q \in \mathbb{Q}^+$q∈Q+ forms a countable basis for the topology, as every open set is a union of such balls and the set is countable.^[10] Conversely, if $X$X is second countable with countable basis $\{U_n \mid n \in \mathbb{N}\}${Un​∣n∈N}, select a point $y_n \in U_n$yn​∈Un​ for each $n$n; the set $\{y_n\}${yn​} is countable and dense, since every nonempty open set contains some $U_n$Un​ and thus intersects $\{y_n\}${yn​}.^[10] The Urysohn metrization theorem reinforces this alignment: every second countable regular Hausdorff space is metrizable, implying that separable metric spaces inherit second countability as a defining feature under regularity.^[20] Representative examples include the real line $\mathbb{R}$R with its standard metric, which is separable via the dense rationals $\mathbb{Q}$Q and thus second countable, and the sequence spaces $\ell^p$ℓp for $1 \leq p < \infty$1≤p<∞, which are separable Banach spaces with countable dense subsets like rational sequences with finite support.^[21] In contrast to general topological spaces, where the properties may diverge, the metric imposes their coincidence.^[10]

Cardinality and general properties

Cardinality bounds

In separable metric spaces, the cardinality of the underlying set $X$X is at most the cardinality of the continuum $\mathfrak{c} = 2^{\aleph_0}$c=2ℵ0​. To see this, let $D = \{d_n : n \in \mathbb{N}\}$D={dn​:n∈N} be a countable dense subset of $X$X. The map $f: X \to \mathbb{R}^\mathbb{N}$f:X→RN defined by $f(x) = (d(x, d_1), d(x, d_2), \dots)$f(x)=(d(x,d1​),d(x,d2​),…) is injective, since if $f(x) = f(y)$f(x)=f(y), then $x$x and $y$y have the same distances to every point in the dense set $D$D, implying $x = y$x=y by density. As $|\mathbb{R}^\mathbb{N}| = \mathfrak{c}$∣RN∣=c, it follows that $|X| \leq \mathfrak{c}$∣X∣≤c.^[22] The real line $\mathbb{R}$R with the standard topology achieves this bound, having cardinality $\mathfrak{c}$c while being separable via the rationals $\mathbb{Q}$Q.^[22] In regular separable topological spaces (without assuming metrizability), the cardinality is at most $|X| \leq 2^\mathfrak{c}$∣X∣≤2c. This follows from separability implying density character $\mathrm{dens}(X) = \aleph_0$dens(X)=ℵ0​, and for regular spaces the weight satisfies $w(X) \leq 2^{\mathrm{dens}(X)} = \mathfrak{c}$w(X)≤2dens(X)=c. Since any topological space satisfies $|X| \leq 2^{w(X)}$∣X∣≤2w(X) (as points are separated by their neighborhood filters into the power set of a base of size $w(X)$w(X)), the bound holds. Without the regularity assumption, even in Hausdorff spaces, larger cardinalities up to $2^{2^{2^\mathfrak{c}}}$222c are possible.^[23][24] This bound is sharp: the Stone-Čech compactification $\beta \mathbb{N}$βN of the natural numbers has cardinality $2^\mathfrak{c}$2c and is separable, as $\mathbb{N}$N embeds densely into it and is countable.^[25] Separability also imposes restrictions on certain point sets. For instance, the set of isolated points in a separable space has cardinality at most $\aleph_0$ℵ0​. If $p \in X$p∈X is isolated, then $\{p\}${p} is open, so any dense subset $D$D must intersect $\{p\}${p}, implying $p \in D$p∈D. Thus, all isolated points lie in the countable dense set.^[22]

Embedding properties

A fundamental embedding property of separable metric spaces is their ability to be completed to a complete separable metric space. Specifically, every separable metric space $(X, d)$(X,d) admits an isometric embedding into its metric completion $\hat{X}$X^, which is a complete metric space, and the image of $X$X is dense in $\hat{X}$X^; moreover, since $X$X has a countable dense subset, so does $\hat{X}$X^.^[26] This completion process preserves the separability while adding completeness, allowing the study of Cauchy sequences and limits within a nicer ambient space. Another key result is the homeomorphic embedding of any separable metric space into the Hilbert cube $[0,1]^\mathbb{N}$[0,1]N, which is itself a compact, complete, separable metric space. This embedding is constructed using a countable basis $\{U_n\}_{n \in \mathbb{N}}${Un​}n∈N​ for the topology of $X$X, defining continuous functions $f_n: X \to [0,1]$fn​:X→[0,1] by $f_n(x) = d(x, X \setminus U_n)$fn​(x)=d(x,X∖Un​), and mapping $x \mapsto (f_n(x))_{n \in \mathbb{N}}$x↦(fn​(x))n∈N​; the resulting map is a homeomorphism onto its image.^[27] The Hilbert cube thus acts as a universal target space, containing homeomorphic copies of all separable metric spaces as subspaces. Separable completely metrizable spaces, known as Polish spaces, further exemplify these properties. Every separable metric space embeds homeomorphically into a Polish space, typically via its completion, which equips the space with a compatible complete metric while retaining separability. Polish spaces are particularly useful in descriptive set theory and analysis due to their rich structure. In the broader context of non-metric separable topological spaces, the countable dense subset inherent to separability allows such spaces to be viewed as subspaces of more structured environments, though embeddings into complete metric spaces generally require additional conditions like regularity and second countability. However, the metric case remains the primary setting where these embedding theorems apply robustly.

Special topics

Constructive mathematics

In constructive mathematics, particularly in Errett Bishop's framework, the real line $\mathbb{R}$R is separable, with the rationals $\mathbb{Q}$Q forming a countable dense subset. This density is proved constructively: for any real $x$x and positive rational $\epsilon$ϵ, there exists a rational $q$q such that $|x - q| < \epsilon$∣x−q∣<ϵ, constructed explicitly from the Cauchy sequence defining $x$x with its modulus of convergence.^[28] Unlike classical mathematics, where equality of reals is decidable via LEM, constructive reals lack decidable equality, but this does not affect the proof of density. Bishop's approach emphasizes "completely presented" sets, where objects are given by explicit constructions allowing decidable membership and equality. For example, the rationals are completely presented, enabling effective approximations. In constructive topology, spaces like the reals are often treated as countably based metric spaces, where the basis of open intervals with rational endpoints is enumerable, supporting separability without non-effective choices.^[28] The distinction from classical separability lies in the requirement for all proofs to provide explicit constructions, avoiding non-constructive existence principles. For instance, while classical proofs may use LEM implicitly, Bishop-style constructive analysis proves key results like completeness and separability using only computable methods, underpinning theorems such as uniform continuity on compact sets. This framework aligns with algorithmic verifiability, treating reals via Cauchy sequences of rationals with explicit moduli, ensuring effective approximations without a "weakened" sense of separability—the property holds fully constructively.^[28] Note that other schools of constructive mathematics, such as Brouwer's intuitionism, may handle topological properties differently, potentially requiring additional principles for certain results.

Further examples

The space of continuous real-valued functions on the compact interval $[0,1]$[0,1], denoted $C[0,1]$C[0,1] and equipped with the supremum norm $\|f\|_\infty = \sup_{x \in [0,1]} |f(x)|$∥f∥∞​=supx∈[0,1]​∣f(x)∣, is a separable metric space. By the Weierstrass approximation theorem, any continuous function on $[0,1]$[0,1] can be uniformly approximated by polynomials; moreover, the subset consisting of polynomials with rational coefficients forms a countable dense subset, establishing separability.^[29] In probability theory, the standard Borel space on $[0,1]$[0,1] — comprising the interval $[0,1]$[0,1] with its Borel $\sigma$σ-algebra — is a separable metric space, as $[0,1]$[0,1] is a Polish space and its Borel $\sigma$σ-algebra inherits the separability properties of the underlying metric structure.^[30] This separability facilitates the study of measurable functions and stochastic processes on such spaces, ensuring the existence of countable dense sets for integration and convergence arguments. A contrasting advanced example is the space of all Radon measures on $[0,1]$[0,1], which serves as the continuous dual of $C[0,1]$C[0,1] and can be equipped with the total variation norm. This space is non-separable, as the set of Dirac delta measures $\{\delta_x \mid x \in [0,1]\}${δx​∣x∈[0,1]} forms an uncountable discrete subset under this norm — with $\|\delta_x - \delta_y\|_1 = 2$∥δx​−δy​∥1​=2 for distinct $x, y$x,y — necessitating an uncountable dense subset to approximate all such points.^[31] In functional analysis, the separability of a Hilbert space guarantees the existence of a countable orthonormal basis, allowing elements to be represented as infinite linear combinations with square-summable coefficients and enabling applications like Fourier series expansions.^[32] Conversely, non-separability arises in uncountable products of separable spaces, such as the product $\prod_{i \in I} [0,1]$∏i∈I​[0,1] where $|I| > \aleph_0$∣I∣>ℵ0​; this space is compact by Tychonoff's theorem but non-separable, highlighting how separability fails under uncountable products and restricting properties like metrizability despite compactness.^[33]