The boundary of a subset of Euclidean space has empty interior, and furthermore has Lebesgue measure zero.Well,this is generally not true,but I can't find an explicit counter-example right now.
Similarly,in set topology.Please show that there exists a metric space,such that the closure of the open ball of radius r, is not the closed ball of radius r in that metric space
Motivation:Actually,there are many such kinds of examples in measure theory and set topology which are not consistent with our intuitions.I find that clarifying these fuzzy definitions or false believes may sometimes be very helpfui.