# Is there such a thing as the standard metric on $\mathbb{R}^N$?

I've recently seen the Heine–Borel theorem stated as "In $\mathbb{R}^N$ with the standard metric, a set $k$ is compact if and only if it is closed and bounded" however I've never heard of "the standard metric on $\mathbb{R}^N$". Is this simply a typo or is there a convention than I'm unaware of?

• Did you try to search yourself? This is explained in every text on topology. Also in online-dictionaries, like here. – Dietrich Burde Dec 20 '17 at 10:50

The standard norm is the Euclidean norm $\|x\|:=\sqrt{\sum_{i=1}^Nx_i^2}$ while the standard metric is induced by the Euclidean norm, so $d(x,y)=\|x-y\|$.