The rational numbers have a problem that makes them unsuitable for use in calculus — they have "gaps" in them\. This may not be obvious or even make sense at first, because the rational numbers are dense in themselves — between any two rational numbers you can always find infinitely many other rational numbers\. How could there be gaps in a set like that? $~$\\newcommand{\\rats}{\\mathbb{Q}} \\newcommand{\\Ql}{\\rats^\\le} \\newcommand{\\Qr}{\\rats^\\ge} \\newcommand{\\Qls}{\\rats^<} \\newcommand{\\Qrs}{\\rats^>}$~$ $~$\\newcommand{\\set}\[1\]{\\left\\{\#1\\right\\}} \\newcommand{\\sothat}{\\ |\\ }$~$

I think that every metric space is dense in itself. If X is a metric space, then a set E is dense in X whenever every element of X is either a limit point of E *or an element of E* (or both).