Let $(X, d)$ be a non-empty metric space (i.e. $X \not = \emptyset$). As is well known, $X$ is called separable if it contains a countable, dense subset.
If the non-empty metric space $(X,d)$ is separable, then a simple consequence is that every (non-empty) subspace $U \subseteq X$ is separable as well (see for example the first property here). I was wandering if the converse statement is true as well:
Is a (non-empty) metric space separable if every subspace of it is separable?
I couldn't find anything about this question on the internet, please excuse if this is a somewhat "easy" or "trivial" question. Thanks for any help, advice, counterexample etc.!
Edit: As stated in the comments by Cameron Williams and David C. Ullrich, the question should have been formulated as follows:
Is a (non-empty) metric space separable if every proper subspace of it is separable?