# Separable space

In topology, a **separable space** is a topological space that contains a countable dense subset, that is, a countable set of points whose closure is the whole space<sup>[1](https://www.math.ucsd.edu/~bdriver/240A-C-03-04/Lecture_Notes/Older-Versions/part3.pdf)</sup>. Separability is one of the countability axioms: it restricts how large a space must be to distinguish its open sets, and it is a standard working hypothesis in analysis because many theorems about metric spaces and Banach spaces hold only under it<sup>[2](https://jiahaotian-stat.github.io/Notes/en/Topology/12/)</sup>.

| Key fact | Detail |
|---|---|
| Definition | A space is separable if it has a countable dense subset, a countable set whose closure is the whole space<sup>[1](https://www.math.ucsd.edu/~bdriver/240A-C-03-04/Lecture_Notes/Older-Versions/part3.pdf)</sup> |
| Basic example | The rationals ℚ<sup>d</sup> are a countable dense subset of ℝ<sup>d</sup><sup> • </sup><sup>[1](https://www.math.ucsd.edu/~bdriver/240A-C-03-04/Lecture_Notes/Older-Versions/part3.pdf)</sup> |
| Metric case | For a metric space, separability, second-countability, and the Lindelöf property are equivalent<sup>[3](https://www.math.ucla.edu/~sharifi/notes/topology-ch04.html)</sup> |
| Heredity | Every subset of a separable metric space is separable in the induced topology<sup>[1](https://www.math.ucsd.edu/~bdriver/240A-C-03-04/Lecture_Notes/Older-Versions/part3.pdf)</sup> |
| Products | Countable products of separable spaces are separable; arbitrary products need not be, but a product of up to continuum many separable spaces is separable (Hewitt–Marczewski–Pondiczery theorem)<sup>[4](https://ncatlab.org/nlab/show/separable%2Bspace)</sup> |
| Standard counterexample | The Sorgenfrey line ℝ<sub>l</sub> is separable but not second-countable; its square is separable yet contains an uncountable discrete, hence nonseparable, subspace<sup>[4](https://ncatlab.org/nlab/show/separable%2Bspace)</sup> |

## Definition and first examples

A subset D of a topological space X is dense if its closure equals X, written \( \overline{D} = X \); equivalently, D meets every nonempty open set<sup>[3](https://www.math.ucla.edu/~sharifi/notes/topology-ch04.html)</sup>. The space is separable when some such D is countable<sup>[1](https://www.math.ucsd.edu/~bdriver/240A-C-03-04/Lecture_Notes/Older-Versions/part3.pdf)</sup>.

The prototype is the real line: the rational numbers ℚ are countable and dense in ℝ with the usual topology, so ℝ is separable<sup>[1](https://www.math.ucsd.edu/~bdriver/240A-C-03-04/Lecture_Notes/Older-Versions/part3.pdf)</sup>. The same holds in every finite dimension, since the set ℚ<sup>d</sup> of rational-coordinate points is countable and dense in ℝ<sup>d</sup><sup> • </sup><sup>[1](https://www.math.ucsd.edu/~bdriver/240A-C-03-04/Lecture_Notes/Older-Versions/part3.pdf)</sup>. In function spaces, the same idea gives separability of the spaces \( c_p(\mu) \) for \( 1 \le p < \infty \), using finite-support vectors with coordinates drawn from a countable dense set of the scalar field<sup>[1](https://www.math.ucsd.edu/~bdriver/240A-C-03-04/Lecture_Notes/Older-Versions/part3.pdf)</sup>.

At the other extreme, discreteness measures nonseparability. In a [discrete space](https://www.edgepedia.org/discrete-space) the only dense subset is the whole space, so a discrete space is separable exactly when it is countable; uncountable discrete spaces are not separable, and are also neither second-countable, Lindelöf, nor σ-compact<sup>[5](https://web.math.ucsb.edu/~bigelow/books/counterexamples.pdf)</sup>. Indiscrete spaces sit at the opposite pole: any nonempty set is dense in the indiscrete topology, so every indiscrete space is separable<sup>[6](https://www.uvm.edu/~smillere/Topology-notes.pdf)</sup>.

## Relation to the countability axioms

Separability sits among several countability conditions that are logically distinct in general topology but coincide for metric spaces.

**Second-countability implies separability.** If a space has a countable base, choosing one point from each basic open set produces a countable dense subset<sup>[3](https://www.math.ucla.edu/~sharifi/notes/topology-ch04.html)</sup>. Second-countable spaces are also Lindelöf, meaning every [open cover](https://www.edgepedia.org/open-cover) has a countable subcover<sup>[3](https://www.math.ucla.edu/~sharifi/notes/topology-ch04.html)</sup>.

**The converse fails.** The real line with the half-open interval topology, denoted ℝ<sub>l</sub> (the Sorgenfrey line), is separable and Lindelöf, and even first-countable, but it is not second-countable: any base would have to be uncountable<sup>[4](https://ncatlab.org/nlab/show/separable%2Bspace)</sup><sup> • </sup><sup>[2](https://jiahaotian-stat.github.io/Notes/en/Topology/12/)</sup>. So separability is strictly weaker than second-countability, and first-countability together with separability still does not imply second-countability<sup>[4](https://ncatlab.org/nlab/show/separable%2Bspace)</sup>.

**For metric spaces the three properties coincide.** A theorem of metric-space topology states that for a metric space X the following are equivalent: X is second-countable, X is Lindelöf, and X is separable<sup>[3](https://www.math.ucla.edu/~sharifi/notes/topology-ch04.html)</sup>. The key step is that a countable dense set S yields a countable base consisting of the balls \( B(y, 1/n) \) with \( y \in S \) and \( n \ge 1 \)<sup>[3](https://www.math.ucla.edu/~sharifi/notes/topology-ch04.html)</sup>.

**Heredity.** In general topology, separability is not hereditary to all subspaces. In metric spaces, however, every subset Y of a separable metric space is separable in the induced topology<sup>[1](https://www.math.ucsd.edu/~bdriver/240A-C-03-04/Lecture_Notes/Older-Versions/part3.pdf)</sup>; equivalently, second-countability is hereditary, and so separability and the Lindelöf property are hereditary in metric spaces<sup>[7](https://garden.jerrylicious.me/t/MW)</sup>.

## Products and counterexamples

Products behave well in the countable case: if each \( X_k \) is separable with countable dense subset \( B_k \), the product \( \prod_{k=1}^{\infty} X_k \) with the product topology is separable<sup>[8](https://androma.org/theorems/946)</sup>. The argument requires some care because the full product \( \prod_n B_n \) of the dense sets is generally uncountable; one works instead with points that differ from a fixed point in only finitely many coordinates<sup>[9](https://math.stackexchange.com/questions/1679691/a-countable-product-of-separable-spaces-is-separable?noredirect=1)</sup>.

Arbitrary products are a different matter. An arbitrary product of separable spaces need not be separable, but the Hewitt–Marczewski–Pondiczery theorem shows that a product of as many as continuum many separable spaces is still separable<sup>[4](https://ncatlab.org/nlab/show/separable%2Bspace)</sup>. More precisely, a product of non-trivial separable spaces is separable if and only if the number of factors is at most \( 2^{\omega} = \mathfrak{c} \), the cardinality of the continuum<sup>[10](https://math.stackexchange.com/questions/526454/uncountable-product-of-separable-spaces-is-separable)</sup>.

The Sorgenfrey plane shows that separability can be lost by subspaces even in a product of two separable lines. The space \( \mathbb{R}_l \times \mathbb{R}_l \) is separable, but the subspace defined by the equation \( y = -x \) is uncountable and discrete, and therefore not separable<sup>[4](https://ncatlab.org/nlab/show/separable%2Bspace)</sup>. This is the standard example separating separability from hereditary separability outside metric spaces.

## Separability in analysis

In functional analysis, separability of a [Banach space](https://www.edgepedia.org/banach-space), a complete [normed vector space](https://www.edgepedia.org/normed-vector-space), is a standing hypothesis for many structure theorems.

Separability also generates open problems. The separable quotient problem, posed by Stefan Mazur in 1932, asks whether every infinite-dimensional Banach space has an infinite-dimensional separable quotient; it remains open<sup>[11](https://arxiv.org/abs/1709.09646)</sup><sup> • </sup><sup>[12](https://doi.org/10.1016/j.topol.2013.07.048)</sup>. All reflexive Banach spaces, and more generally all infinite-dimensional weakly compactly generated Banach spaces, have separable quotients<sup>[11](https://arxiv.org/abs/1709.09646)</sup>. Related classical questions include the Suslin Problem, which asked whether there is a nonseparable linearly ordered space in which families of pairwise disjoint open intervals are at most countable<sup>[12](https://doi.org/10.1016/j.topol.2013.07.048)</sup>.

Weak forms of separability have their own calculus: a space that is nowhere-dense separable (nwd-separable) has separable finite products with any space, and set-theoretic constructions such as the Ellentuck topology show that this property can fail in ways ordinary separability does not<sup>[12](https://doi.org/10.1016/j.topol.2013.07.048)</sup>.

## Why the countable dense set matters

A countable dense subset lets arguments pass from uncountable spaces to countable ones. This is the mechanism behind the equivalence with second-countability and the Lindelöf property, and it underlies the Urysohn metrization theorem, which identifies the second-countable regular Hausdorff spaces with subspaces of the countable power \( [0,1]^{\mathbb{N}} \)<sup>[3](https://www.math.ucla.edu/~sharifi/notes/topology-ch04.html)</sup>.

## References

1. [Topological Spaces, lecture notes (Driver, UCSD), Definitions 10.40–10.42](https://www.math.ucsd.edu/~bdriver/240A-C-03-04/Lecture_Notes/Older-Versions/part3.pdf)
2. [Point-Set Topology, Chapter 4: Countability and separation axioms (Romyar Sharifi, UCLA)](https://www.math.ucla.edu/~sharifi/notes/topology-ch04.html)
3. [Ch 12: Countability, topology notes](https://jiahaotian-stat.github.io/Notes/en/Topology/12/)
4. [Separable space, nLab](https://ncatlab.org/nlab/show/separable%2Bspace)
5. [Counterexamples (Bigelow, UCSB), §42](https://web.math.ucsb.edu/~bigelow/books/counterexamples.pdf)
6. [Separable spaces, topology notes (Miller, UVM)](https://www.uvm.edu/~smillere/Topology-notes.pdf)
7. [Proposition 8.1.2, Jerry's Digital Garden](https://garden.jerrylicious.me/t/MW)
8. [Countable Products of Separable Spaces, Theorem 946, Androma](https://androma.org/theorems/946)
9. [A countable product of separable spaces is separable, Mathematics Stack Exchange](https://math.stackexchange.com/questions/1679691/a-countable-product-of-separable-spaces-is-separable?noredirect=1)
10. [Uncountable product of separable spaces is separable?, Mathematics Stack Exchange](https://math.stackexchange.com/questions/526454/uncountable-product-of-separable-spaces-is-separable)
11. [On the separable quotient problem for Banach spaces, arXiv:1709.09646](https://arxiv.org/abs/1709.09646)
12. [Comparing weak versions of separability, Topology and its Applications](https://doi.org/10.1016/j.topol.2013.07.048)

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