To be specific, a topological space is separable if and only if it has a subset that is both countable and dense.
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 the only sorts of proofs acceptable in constructive analysis. A famous example of a theorem of this sort is the HahnBanach theorem.
Every second countable space is separable. As a partial converse, every separable metric space must be second countable. More generally, every separable uniform space whose uniformity has a countable basis must be second countable.
An example of a separable space that is not second countable is R_{llt}, the set of real numbers equipped with the lower limit topology[?]. To avoid violating the previous paragraph, it follows that R_{llt} must not be metrisable  it can't be made into a metric space. On the other hand, because R_{llt} is completely regular, it is uniformisable  it can be made into a uniform space. But again, to avoid violating the previous paragraph, none of its uniformities could possibly have a countable basis.
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