Pretopology

Pretopology is a branch of mathematics that studies pretopological spaces. A pretopological space is a set equipped with a notion of "closure" that does not necessarily satisfy all the axioms of a topological space. Specifically, in a pretopological space, the closure operator can be reflexive and monotonic, but it is not required to be idempotent or to satisfy the empty set axiom.

Formally, a pretopology on a set X is a function c: P(X) -> P(X), where P(X) is the power set of X, satisfying:

1. x in c({x}) for all x in X (reflexivity)

2. If A is a subset of B, then c(A) is a subset of c(B) (monotonicity)

In a topological space, the closure operator also satisfies:

3. c(empty set) = empty set

4. c(A union B) = c(A) union c(B)

5. c(c(A)) = c(A) (idempotency)

Pretopological spaces relax these latter two axioms. The study of pretopologies is motivated by the desire to capture notions of proximity and convergence in a more general framework than standard topology. This can be useful in areas such as fuzzy sets, rough sets, and certain aspects of theoretical computer science.

A pretopology can be converted into a topology by imposing additional conditions. For instance, if the closure operator is idempotent, then the pretopological space is a topological space. Various ways exist to derive a topology from a pretopology, often by modifying the closure operator or by defining a derived open set system. The relationship between pretopologies and topologies allows for a richer understanding of topological concepts by examining their more generalized forms.