In a metric space
, a set is closed
iff every convergent sequence in the set has its limit
point contained within the set.
The topological space definition given above is used because it is true for all metric spaces:
Suppose U is a open subset of X, with complement V. Then a convergent sequence (xn) all of whose terms are contained in V cannot converge to x in U, because there exists some positive ε such that B(x,ε) is contained in U which must also contain some (in fact, infinitely many) terms of the sequence, giving a contracdiction. Therefore x is in V and V is closed.
Conversely, if U is not open, then there exists some point x in U such that B(x,1/n) is not contained in U for any n. Then the ball has nonempty intersection with the complement V (which is nonempty) for all n, so can find some point xn in V for each n, which defines a sequence in V converging to x, which is not in V, so V is not closed.