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?

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    $\begingroup$ Did you try to search yourself? This is explained in every text on topology. Also in online-dictionaries, like here. $\endgroup$ – 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 $d(x,y)=\|x-y\|$.

  • $\begingroup$ It even doesn't depend on the given norm since they are all equivalent in the finite case. $\endgroup$ – Michael Hoppe Dec 20 '17 at 11:06
  • $\begingroup$ The norms are equivalent but not equal and same goes for the metric.Since they induces the same topology, the Heine-Borel theorem works for all norms and all metrics induced by a norm. But the question was about the standard metric, which is the Euclidean distance. $\endgroup$ – Mundron Schmidt Dec 20 '17 at 11:17

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