[K4] Finally, fix formula_98. Axiom [T2] implies formula_99; furthermore, axiom [T2] implies that formula_100. By extensivity [K2] one has formula_101 and formula_102, so that formula_103. But formula_104, so that all in all formula_105. Since then formula_106 is a minimal element of formula_107 w.r.t. inclusion, we find formula_108. Point 4. ensures additivity [K4]. In fact, these two complementary constructions are inverse to one another: if formula_109 is the collection of all Kuratowski closure operators on formula_1, and formula_111 is the collection of all families consisting of complements of all sets in a topology, i.e. the collection of all families satisfying [T1]—[T3], then formula_112 such that formula_113 is a bijection, whose inverse is given by the assignment formula_114. First we prove that formula_115, the identity operator on formula_109. For a given Kuratowski closure formula_117, define formula_118; then if formula_73 its primed closure formula_120 is the intersection of all formula_4-stable sets that contain formula_75. Its non-primed closure formula_123 satisfies this description: by extensivity [K2] we have formula_124, and by idempotence [K3] we have formula_125, and thus formula_126. Now, let formula_127 such that formula_128: by isotonicity [K4'] we have formula_129, and since formula_130 we conclude that formula_131. Hence formula_123 is the minimal element of formula_133 w.r.t. inclusion, implying formula_134. Now we prove that formula_135. If formula_136 and formula_137 is the family of all sets that are stable under formula_77, the result follows if both formula_139 and formula_140. Let formula_141: hence formula_142. Since formula_89 is the intersection of an arbitrary subfamily of formula_72, and the latter is complete under arbitrary intersections by [T2], then formula_145. Conversely, if formula_146, then formula_89 is the minimal superset of formula_75 that is contained in formula_72. But that is trivially formula_75 itself, implying formula_141.