topology. Properties Since any Kuratowski closure is isotonic, and so is obviously any inclusion mapping, one has the (isotonic) Galois connection , provided one views as a poset with respect to inclusion, and as a subposet of . Indeed, it can be easily verified that, for all and , if and only if . If is a subfamily of , then If , then . Topological concepts in terms of closure Refinements and subspaces A pair of Kuratowski closures such that for all induce topologies such that , and vice versa. In other words, dominates if and only if the topology induced by the latter is a refinement of the topology induced by the former, or equivalently . For example, clearly dominates (the latter just being the identity on ). Since the same conclusion can be reached substituting with the family containing the complements of all its members, if is endowed with the partial order for all and is endowed with the refinement order, then we may conclude that is an antitonic mapping between posets. In any induced topology (relative to the subset A) the closed sets induce a new closure operator that is just the original closure operator restricted to A: , for all . Continuous maps, closed maps and homeomorphisms A function is continuous at a point iff , and it is continuous everywhere iff for all subsets . The mapping is a closed map iff the reverse inclusion holds, and it is a homeomorphism iff it is both continuous and closed, i.e. iff equality holds. Separation axioms Let be a Kuratowski closure space. Then is a T0-space iff implies ; is a T1-space iff for all ; is a T2-space iff implies that there exists a set such that both and , where is the set complement operator. Closeness and separation A point is close to a subset if This can be used to define a proximity relation on the points and subsets of a set. Two sets are separated iff . The space is connected iff it cannot be written as the union of two separated subsets. See also Notes References . . . . . External links Alternative Characterizations of Topological Spaces Closure operators Mathematical axioms