Prove that seperability is a topological property
Suppose that is a separable space and a topological space with a homeomorphism between them. Let us prove that is separable.
As is separable, there exists a countable dense subset . We claim that is a countable dense subset of First of all, as is a homeomorphism, it is, in particular, a bijection, so is countable. Secondly, we remark that for a continuous function we have for any . Indeed, as is continuous, is a closed set containing , it contains, in particular, its closure. Finally, we apply this as following : , but (as is bijective, ) , so and therefore . Applying to both parts, remembering that it is bijective, we conclude that . Therefore, is a countable dense subset of , so it is separable.
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