Inscrit le: 07 Oct 2017
|Posté le: Mer 13 Déc - 09:34 (2017) Sujet du message: Separable space
We have seen the concept of separable space in progress: A space is separable if it contains a dense part at most countable. My question is: If a topological space is separable, are all its topological subspaces also separable?
I think so, but I do not know how to prove it If Morphism, Prauron or Klaus are around ... The basic idea for your first question is that a compact space is always separable.
So you take a non-separable space and you take a compactifié, it gives you a compact space so separable that contains a non-separable subspace. Hence my question: And if we want a noncompact space?
In passing, I specify that if we add "metric" to our assumptions, then we can no longer find an example: Any subspace of a separable metric is still separable, it is not too difficult to prove with a distance at hand.
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