Separable Space

A separable space is a topological space containing a countable dense subset. Although the space itself may have uncountably many points, a countable collection meets every nonempty open region. Separability is a countability condition in topology, with particularly strong consequences for metric spaces. (math.ucla.edu)

Definition and interpretation

A space $X$ is separable if there exists a countable subset $D\subseteq X$ such that

where $\overline{D}$ denotes the closure of $D$. Equivalently, every nonempty open set in $X$ intersects $D$. “Countable” here includes finite sets, so every countable topological space is separable: the whole space is a countable dense subset. (mathweb.ucsd.edu)

In a metric space $(X,d)$, the condition becomes

Thus each point can be approximated arbitrarily closely by points from one fixed countable set. In particular, choosing $a_n\in D$ with $d(x,a_n)<1/n$ gives a sequence converging to $x$. This sequential interpretation uses the metric; the general definition is formulated using closure and open sets. (stat.umn.edu)

For example, the rational numbers $\mathbb Q$ are dense in the real line $\mathbb R$. Consequently, $\mathbb R$ is separable despite being uncountable. More generally, $\mathbb Q^n$ is a countable dense subset of Euclidean space $\mathbb R^n$. (stat.umn.edu)

Relations to other countability conditions

A second-countable space has a countable basis for its topology. Every such space is separable: select one point from each nonempty basic open set. The selected points form a countable set meeting every nonempty open set. The converse fails for general topological spaces. (math.ucla.edu)

For metric spaces, however, the following conditions are equivalent:

• separability;
• second countability;
• the Lindelöf property, meaning that every open cover has a countable subcover. (math.ucla.edu)

The central construction is explicit. If $D$ is countable and dense, then

is a countable basis, where $B(a,r)$ is an open ball. Given $x\in U$ with $U$ open, choose $\varepsilon>0$ such that $B(x,\varepsilon)\subseteq U$. Take $a\in D$ with $d(x,a)<\varepsilon/3$, and a rational radius satisfying

The triangle inequality then gives

Thus the countable family describes the entire topology. (mssc.mu.edu)

Preservation under constructions

Separability is preserved by several standard operations:

• Continuous images. If $f:X\to Y$ is continuous and $X$ is separable, its image $f(X)$, with the subspace topology, is separable. The image of a countable dense set is dense in $f(X)$.
• Quotients. A space with the quotient topology obtained from a separable space is separable, since the quotient map is continuous and surjective.
• Countable products. A finite or countable product of separable spaces is separable in the product topology. (math.wvu.edu)

For the countable-product assertion, choose countable dense subsets $D_n\subseteq X_n$ and fixed points $b_n\in D_n$. Tuples whose coordinates belong to $D_n$ and equal $b_n$ except at finitely many positions form a countable dense subset. Density follows because a basic product-open set restricts only finitely many coordinates. The topology matters: this argument does not apply to arbitrary alternative topologies on the same product. (math.wvu.edu)

Every subspace of a separable metric space is separable. One can restrict a countable topological basis to the subspace and select points from its nonempty members. Simply intersecting the original dense set with the subspace need not work: $\mathbb Q$ is dense in $\mathbb R$, but its intersection with the irrational numbers is empty. (mathweb.ucsd.edu)

Nonmetrizable examples and limitations

The Sorgenfrey line is $\mathbb R$ with a topology generated by half-open intervals $[a,b)$. It is separable because every nonempty basic interval contains a rational number, but it is not second-countable. Thus separability alone does not guarantee metrizability or a countable topological basis. (math.wvu.edu)

Separability also need not pass to subspaces outside the metric setting. The Sorgenfrey plane, the product of two Sorgenfrey lines, is separable, but its anti-diagonal

is an uncountable discrete subspace. Indeed,

meets $A$ only at $(x,-x)$. A discrete space has no proper dense subset, so this subspace is not separable. (math.wvu.edu)

Separability should therefore be distinguished from hereditary separability, the stronger requirement that every subspace be separable. Separable metric spaces satisfy the stronger condition, whereas general separable spaces need not. (mathweb.ucsd.edu)

Function spaces and Hilbert spaces

Separability is important in functional analysis, where points of a space may themselves be sequences or functions. For $1\leq p<\infty$, the sequence spaces $\ell^p$, members of the family of $L^p$ spaces, are separable. Finitely supported sequences with rational coordinates—or rational real and imaginary parts in the complex case—form countable dense subsets. (mathweb.ucsd.edu)

By contrast, $\ell^\infty$, the space of bounded sequences with the supremum norm, is not separable. Its uncountable set of binary sequences has pairwise distance $1$. Their open balls of radius $1/3$ are disjoint, and any dense subset must meet each ball, requiring uncountably many points. More generally, an uncountable uniformly separated set prevents separability in a metric space. (mathweb.ucsd.edu)

A Hilbert space is separable exactly when it admits a finite or countably infinite orthonormal basis. Starting from a countable dense set, the Gram–Schmidt process, with dependent vectors omitted, produces such a basis. Conversely, finite linear combinations of basis vectors with rational coefficients form a countable dense set. (mathweb.ucsd.edu)

For a countably infinite orthonormal basis $(e_n)$, vectors have norm-convergent expansions

This is a topological expansion, not an algebraic basis representation: an algebraic basis requires finite linear combinations. Separability supplies countable coordinates without requiring the space to be finite-dimensional. (mathweb.ucsd.edu)

Completeness and Polish spaces

Separability and completeness impose different conditions. Separability concerns a countable dense subset; completeness requires every Cauchy sequence to converge within the space. The rational numbers with their usual metric are separable but incomplete, while an uncountable set with the discrete metric is complete but nonseparable. (stat.umn.edu)

A Polish space is a separable topological space admitting a compatible complete metric. The existence of such a metric matters, rather than completeness of every compatible metric. Euclidean spaces and separable Banach spaces are examples. Polish spaces provide a standard setting for probability and measure theory, where countable topological structure and completeness are used together. (mat.univie.ac.at)

References

1. Point-Set Topology — Chapter 4 Countability and separation axioms · Romyar Sharifimath.ucla.edu
2. Point-Set Topologymath.ucla.edu
3. Topology course textmssc.mu.edu
4. Analysis lecture notesmathweb.ucsd.edu
5. 3 Polish Spacesstat.umn.edu
6. Lecture notes: Advanced Probabilitymat.univie.ac.at