A closure operator is a mathematical concept used in various fields, including set theory and topology. It is a function that takes a set and returns a closed set, which contains all the limit points of the original set. This operator helps in understanding how sets can be expanded to include their boundary points, making them "closed" in a certain sense. In the context of lattices, a closure operator satisfies specific properties: it is extensive (the original set is included in its closure), idempotent (applying it multiple times does not change the result), and monotonic (if one set is a subset of another, their closures maintain this relationship). These properties make closure operators useful in various mathematical analyses.