be the collection of all sets that contain (or are empty). Then, the closure of is the whole space ( is the smallest closed set containing ), but every set of the form 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 . 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 . A separable Hausdorff space has cardinality at most , where is the cardinality of the continuum. For this closure is characterized in terms of limits of filter bases: if and , then if and only if there exists a filter base consisting of subsets of that converges to . The cardinality of the set of such filter bases is at most . Moreover, in a Hausdorff space, there is at most one limit to every filter base. Therefore, there is a surjection when The same arguments establish a more general result: suppose that a Hausdorff topological space contains a dense subset of cardinality . Then has cardinality at most and cardinality at most if it is first countable. The product of at most continuum many separable spaces is a separable space . In particular the space of all functions from the real line to itself, endowed with the product topology, is a separable Hausdorff space of cardinality . More generally, if is any infinite cardinal, then a product of at most spaces with dense subsets of size at most has itself a dense subset of size at most (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