Why are closure operators important? This is probably a naive question, but I am not a mathematician so I hope you will excuse me.
Can someone explain in layman's terms why closure operators are important? I have read in Wikipedia that they are important in many fields but they do not explain why. What is important in the three properties they respect? (Idempotence, extensivity and montonicity)
Thanks in advance
 A: The importance is a matter of opinion but there is evidence for this in that closure operators appear in many apparently unrelated fields. This is somewhat similar to how category theory ideas also appear in many apparently unrelated fields. In each field the closure operator appears naturally and later they are seen to have common properties. General results that hold for any closure operator then automatically hold in any particular closure operator. This is an advantage in solving problems that depend on using closure operators. You can build a tool kit of general results. The three properties remind us of the somewhat analogous properties of an equivalence relation. In fact, there is an equivalence closure operator for relations which constructs the reflexive, symmetric, transitive closure of a relation and hence is an equivalence relation. Extensivity is analogous to Reflexivity. Monotonicity is analogous to Transitivity.
The Extensivity property means that the closure always is an extension of what you started with. The Monotonicity property means that the operator respects the inclusion relation. The Idempotenticity of a closure operator is a minimality property. Once you have taken a closure, apply closure again gives no more change. These three properties are common sense properties that a closure operator should have based on intuition from examples like the convex hull operator.
The closure operator is like a completion of partial structures. A real life example might be an image or object is inserted in a kaleidoscope and the result is a symmetric image that we associate with kaeidoscopes. Thus a symmetric structure is constructed from an arbitrary starting object with incomplete structure.
