Separable space

In Mathématiques, and more precisely in Topologie, a separable space is a topological Espace containing a countable subset and dense, i.e. if one can find a Suite whose adherence is equal to entire topological space.

Any separable space '' métrisable '' is a Espace at countable base and thus has with more the Puissance of continuous the. Are of this type the majority of usual spaces. To be at countable base is a property much stronger, and much more interesting, than to be separable.

The assumption of separability is found abundantly in the results of analyzes functional.

A subspace of a separable space is not in general separable. On the other hand, a subspace of a space at countable base is still at countable base. A fortiori, by what precedes, a subspace of a separable space métrisable is still métrisable separable. But it is possible to give a direct demonstration of this second assertion without using separable equivalence between métrisable and at countable base.

If X is a separable space, has while being a subspace, chooses X ^N a dense continuation in X , and supposes that the topology of X is defined by a distance D . For all entireties N and m , let us fix, if there are some, a point has _{N, m} of has with D ( has _{N, m}, X _N) <1/m. Is has a point of has . For $\ epsilon>0$ , by definition of the continuation X , there exists an entirety sufficiently large N , with D ( X _N, has ) < $\ epsilon$ . If m >1/ $\ epsilon$ , the point has _{N, m} is in definite fact. And one has then:

d (a_ {N, m}, a)< \ epsilon+1/m< \ epsilon~

The countable continuation has _{N, m} is in fact dense in has .

Examples

The unit $\ R$ of the real numbers

The unit $\ mathbb {R}$ of the real numbers, provided with its usual topology, is separable because $\ mathbb {Q}$ is dense there and of countable cardinal.

Metric space précompact

All metric Espace Précompact is separable.

There exist very large compact spaces not métrisables but nevertheless separable; it is the case of the Compactifié de Stone-Cech of NR which has even power that the whole of the parts of R .

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Separable space

Examples

The unit \ R of the real numbers

Metric space précompact

The unit $\ R$ of the real numbers