: The relevant text in question reads, "a subset of Euclidean space being ... bounded (that is, all points lie within some fixed distance of each other)."  The assertion "a set X is bounded if all points [in X] lie within some fixed distance of each other" is correct; it is also correct when X happens to be a subset of some larger set (like Euclidean space).  Your paraphrase (edited slightly) is that "a bounded set X in a metric space M is a space where there is at most a fixed distance between any two points" is not particularly close to what is written; it is also not false, at least if read in the natural way so that the second copy of the word "space" refers to the set X rather than to the ambient space M.  I would be happy to discuss alternate phrasings, but the sentence is at risk of becoming unreadable due to its length. (I have responded to the irrelevant personal comments on your talk page.)