are all finite sets closed?
d : a set A is closed if every limit point of A is an element of it. ... If M is a metric space then every subset A={x1,…,xn}⊆M is closed. In fact, if a∉A then d(a,A) is the least of the numbers d(a,x1),…,d(a,xn) thus, d(a,A)>0.