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I have a question? Why is it that every subset of a metric space is a metric space? I mean what if the subset is the empty set, then it can't be a metric space, right? because a metric space is by definition a non-empty set. So would I be correct to say that every non-empty subset of a metric space is a metric space? But i'm confused because since compact sets are metric spaces, and every finite set is a compact set, doesn't that mean that the empty set is compact? Therefore a metric space by definition?

mathsssislife
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  • I guess it's possible, but "my" definition of metric space allows $\emptyset$. Not that it matters. – Randall Jan 20 '19 at 01:35
  • I suppose it depends on your definition of metric space. The definition that I am familiar with allows the empty set. Either way I think this is not an important question to ask since the empty set is not an interesting space... – angryavian Jan 20 '19 at 01:36
  • I will always laugh at whoever it was here that had that story from their grad school years about "the keeper of the empty set." – Randall Jan 20 '19 at 01:37
  • @angryavian what if I wanted to prove that a subset of a metric space has a following property that for every element in the set P(x), how would I do that if the empty set is a subset? would I be incorrect to say that if I have something that said Let S $\subset$ X. Prove $\forall x \in S$ P(x), then I must only consider the non- empty sets? or is it still true vacously? what about $\forall x \in S \exists$? or $\exists$..... – mathsssislife Jan 20 '19 at 01:41
  • It's true vacuously, but no one cares. It's not going to suddenly prove some great big theorem because you can apply it to the empty set. – Randall Jan 20 '19 at 01:44
  • @Randall oh so do people usually leave out a proof for the empty set? – mathsssislife Jan 20 '19 at 01:45
  • Usually, yes, because it's trivial and only clutters up your writing. If I'm mature enough to read a paper I don't need an empty argument spelled out for me. – Randall Jan 20 '19 at 01:49
  • Compact sets are not necessarily metric spaces. Even a non-empty compact Hausdorff space may have a topology that cannot be generated by a metric. – DanielWainfleet Jan 20 '19 at 09:43

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I think a metric space can be empty, see Wikipedia.

YuiTo Cheng
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