The derived set of a subset of a space formula_25 need not be closed in general. For example, if formula_26 with the trivial topology, the set formula_27 has derived set formula_28 which is not closed in formula_29 But the derived set of a closed set is always closed. ("Proof:" Assuming formula_1 is a closed subset of formula_31 which shows that formula_19 take the derived set on both sides to get formula_33 i.e., formula_34 is closed in formula_29) In addition, if formula_25 is a T1 space, the derived set of every subset of formula_25 is closed in formula_29