Consequence Operator - Closure Operators in Topology

Closure Operators in Topology

The topological closure of a subset X of a topological space consists of all points y of the space, such that every neighbourhood of y contains a point of X. The function that associates to every subset X its closure is a topological closure operator. Conversely, every topological closure operator on a set gives rise to a topological space whose closed sets are exactly the closed sets with respect to the closure operator.

For topological closure operators the second closure axiom (being increasing) is redundant.