In general topology, a pretopological space is a generalization of the concept of topological space. A pretopological space can be defined in terms of either filters or a preclosure operator. The similar, but more abstract, notion of a Grothendieck pretopology is used to form a Grothendieck topology, and is covered in the article on that topic. Let X be a set. A neighborhood system for a pretopology on X is a collection of filters N(x), one for each element x of X such that every set in N(x) contains x as a member. Each element of N(x) is called a neighborhood of x. A pretopological space is then a set equipped with such a neighborhood system. A net x_{\alpha} converges to a point x in X if x_{\alpha} is eventually in every neighborhood of x. A pretopological space can also be defined as a set X with a preclosure operator (Čech closure operator) The two definitions can be shown to be equivalent as follows: define the closure of a set S in X to be the set of all points x such that some net that converges to x is eventually in S. Then that closure operator can be shown to satisfy the axioms of a preclosure operator. Conversely, let a set S be a neighborhood of x if x is not in the closure of the complement of S. The set of all such neighborhoods can be shown to be a neighborhood system for a pretopology. A pretopological space is a topological space when its closure operator is idempotent. A map between two pretopological spaces is continuous if it satisfies for all subsets