In mathematics, a topological space is called separable if it contains a countable dense subset; that is, there exists a sequence ( x n ) n = 1 ∞ {\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. 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 length- n {\displaystyle n} vectors of rational numbers, r = ( r 1 , … , r n ) ∈ Q n {\displaystyle {\boldsymbol {r}}=(r_{1},\ldots ,r_{n})\in \mathbb {Q} ^{n}} , is a countable dense subset of the set of all length- n {\displaystyle n} vectors of real numbers, R n {\displaystyle \mathbb {R} ^{n}} ; so for every n {\displaystyle n} , 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 { U n } {\displaystyle \{U_{n}\}} is a countable base, choosing any x n ∈ U n {\displaystyle x_{n}\in U_{n}} from the non-empty U n {\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 X {\displaystyle X} , pick some x 0 ∈ X {\displaystyle x_{0}\in X} , and define the topology to be the collection of all sets that contain x 0 {\displaystyle x_{0}} (or are empty). Then, the closure of x 0 {\displaystyle {x_{0}}} is the whole space ( X {\displaystyle X} is the smallest closed set containing x 0 {\displaystyle x_{0}} ), but every set of the form { x 0 , x } {\displaystyle \{x_{0},x\}} is open. Therefore, the space is separable but there cannot have a countable base.
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