# Mistake in Kolmogorov's Elements of the theory of functions and functional analysis?

I was reading the book Elements of the theory of functions and functional analysis (Volume 1) by Kolmogorov. More precisely, the part of compact sets in metric spaces:

The underlined text is so confusig for me. First, we know that in $\mathbb{R}$ the compact sets are the closed and bounded sets (Heine-Borel), then, can be a set that is only bounded compact? I don't think so. I know that the definition that Kolomorov use is, really, the definition of sequentially compact, and I know that the compactness and the sequentially compactness are equivalent in metric spaces. Then, am I losing something of context in the book?

Moreover, the book says that an arbitrary subset of a compact set is compact, but, is it true? I know that $[0,1]\subseteq\mathbb{R}$ is compact but $\left\{\displaystyle\frac{1}{n}:n\in\mathbb{N} \right\}\subseteq[0,1]$ is not compact. Again, Kolmogorov use some definition that I don't have? Am I misunderstanding? I will be really grateful if someone can explain me this. I really appreciate any help you can provide me.

• It looks to me that what Kolmogorov is calling "compact" is in fact what is now more commonly called "precompact" or "relatively compact", meaning that the closure is compact. – carmichael561 Dec 29 '17 at 0:30