Separable Space - Cardinality

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 c. In such a space, closure is determined by limits of sequences and any 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 X.

A separable Hausdorff space has cardinality at most, where c is the cardinality of the continuum. For this closure is characterized in terms of limits of filter bases: if Y is a subset of X and z is a point of X, then z is in the closure of Y if and only if there exists a filter base B consisting of subsets of Y which converges to z. 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 X contains a dense subset of cardinality . Then X 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 (Willard 1970, p. 109, Th 16.4c). 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 κ is any infinite cardinal, then a product of at most 2κ spaces with dense subsets of size at most κ has itself a dense subset of size at most κ (Hewitt–Marczewski–Pondiczery theorem).