How did separable spaces get their name?

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In general topology, how did separable spaces get their name?

It is not intuitively clear to me what can be separated in a separable space, so I was wondering what was the history behind that name.
 
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A lot of people have wondered about this, and I don't think anyone has found a convincing, non-speculative reason. If you search around, you can find that this term was coined by Frechet in his PhD thesis. As such, the question is really "why are metric spaces with a countable dense subset 'separable'?" Maybe because they're second countable?
 

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