The set of the real numbers is closed. For me this is nearly trivial (*) but perhaps I miss something; a colleagues insists that there are some deeper considerations why this is far from trivial - but I don't get his point (*) A) A set is closed if its complement is open; the complement of R is ø which is open, therefore R is closed B) In a topological space, a set is closed if it contains all its limit points; this applies directly to R (see also D) C) In a complete metric space, a set is closed if it's constructed as the closure w.r.t. to its limit operation; the set of real numbers can be constructed as completion of the rational numbers in the sense of equivalence classes of Cauchy sequences; then Q is dense in R by construction and R is closed by construction Do I miss something? Are there more fundamental definitions of closed sets? Is it problematic that closed sets are defined via limiting points of convergend sequences whereas this does not apply to R as a whole b/c it misses the "boundary" of R which could be defined by divergent sequences (xn) → ∞? I do not see such problems, but perhaps I am missing something.