# Subset of metric space

Let $(X,d)$ be a metric space and let $Y$ be a nonempty subset of $X$. Show that $d$ defines a metric space on $Y$.

I'm not sure how to go about this, I was thinking of just checking the properties hold in $Y$, but not sure if that's the correct approach.

Any help is appreciated, thanks.

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You will have to check if the properties hold on $Y$, use that they hold on $X$. – Stefan Oct 17 '12 at 16:07
It is the correct approach. Just verify that the required properties of $d$ also hold in $Y$. – Ayman Hourieh Oct 17 '12 at 16:08
Isn't that trivial though? – Alti Oct 17 '12 at 16:08
Indeed it is trivial. – Ayman Hourieh Oct 17 '12 at 16:15
Thanks for you help! – Alti Oct 17 '12 at 16:17

## 1 Answer

This is indeed the correct approach. Just verify that the required properties of $d$ also hold in $Y$. As you have noticed, it is trivial to do so.

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