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If $(X,d)$ is a metric space and $Y\subseteq X$, does one say $Y$ is a metric space when the metric on $X$, $d$, is restricted to $Y$ or does one say $Y$ is a metric space when $d$ is restricted to $Y\times Y$ since the domain of $d$ is technically the product $X \times X$?

Partey5
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    I usually avoid the issue by saying that $Y$ is a metric space with the metric induced by $d$. Technically this is $d$ restricted to $Y\times Y$, but you’ll often see or hear the sloppier version. – Brian M. Scott Dec 09 '20 at 18:18
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    A metric on $X$ is the function $d : X\times X \to \mathbb{R}_+$ such that... So I would say that the restriction of $d$ on $Y\times Y$ is a metric on $Y$ and call it the induced metric. – Didier Dec 09 '20 at 18:18
  • Thank you for your comments – Partey5 Dec 09 '20 at 20:14
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    I would just say "$Y$ is a metric subspace of $X$", or possibly "$Y$ is a sub metric space of $X$". Intuitively, these imply that the metric is involved in the subspace relatiionship, where the restriction of $d$ is the obvious meaning. Of course when introducing any new terminology, you should define it. But in this case, I consider the likelyhood of mis-understanding by someone who missed the definition to be very small. – Paul Sinclair Dec 10 '20 at 00:53

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