Separable space

The mathematical concept called separable in the topology of a property of spaces that can facilitate reasoning among others. Often you can do without proof techniques such as transfinite induction for sets of such spaces. Spaces with this property are manageable or small, in a way, that is, not boundless great since they can still be treated with countable methods. So you can for example always find countable orthonormal bases in a separable Hilbert space and thus develop each element of the space in a row, that is, countable sum.

Definition

A topological space is called separable if there is a countable subset which is dense in this space.

Criteria for separable spaces

• A topological space with a countable base (second countable ) is separable. This gives the countable dense subset by in the base to pick a point from each set.
• Every compact metrizable space is separable. More precisely, that for metrizable spaces, the three properties zweitabzählbar, lindelöfsch and separable to be equivalent. Compactness is a special case of the Lindelöf property, so that the former statement follows from this equivalence inference.
• A topological vector space (over R or C) is separable if and only if there is a countable subset, so that the vector subspace generated therefrom is tight.

Examples

Examples of separable spaces are approximately:

• The rooms are separable, as is countable and is dense in.
• The spaces Lp () with a limited, open subset and are separable.
• The consequence rooms are separable.
• The space of null sequences is separable with the supremum norm.
• The space of terminating sequences ( ) is the standard for separable.
• The rooms are for natural separable. It refers to an open subset of.

Counter-examples

• The space of bounded sequences is non- separable.
• The space of the quasi- periodic functions is an example of a non- separable Hilbert space.

Permanenzeigenschaften

• Pictures of separable spaces under continuous functions are separable again. As a dense subset in the picture is just the image of the dense subset in the domain.
• Subspaces separable spaces are generally not re- separable, for example, the separable Niemytzki room contains a non - separable subspace, the Sorgenfrey level is another example. It is true, however, that subspaces separable metric spaces are separable again. This follows from the above equivalence of separability and Zweitabzählbarkeit, because the latter carries over obviously on subspaces.
• Is a family of separable spaces and is the cardinality of a maximum equal to the cardinality of the continuum, it is also separable with the product topology. To view this amazing result, it suffices to prove the separability of. To this end, considering it is easy that the countable set of finite sums of functions is dense, the characteristic function of the interval.

Conjunction with other terms

• If the topology of a separable space generated by a metric and is the space with respect to the metric completely, then one speaks of a Polish space.