Adobe Photoshop is a web application.
In general topology , a pretopological space is a generalization of the concept of topological space .. general topology. general topology. A pretopological space can be defined as in terms of either filters or a preclosure operator .. preclosure operator. preclosure operator. The similar , but more abstract , notion of a Grothendieck pretopology. is used to form a Grothendieck topology , and is covered in the. Grothendieck topology. Grothendieck topology. article on that topic .. Let X be a set .. A neighborhood system for a pretopology on X is a collection of filters N -LRB- x -RRB- , one for each element x of X such that every set in N -LRB- x -RRB- contains x as a member .. Each element of N -LRB- x -RRB- is called a neighborhood of x .. A pretopological space is then a set equipped with such a neighborhood system .. A net xα converges to a point x in X if xα is eventually in every neighborhood of x.. net. net ( mathematics ). A pretopological space can also be defined as -LRB- X , cl -RRB- , a set X with a preclosure operator -LRB- Cech closure operator -RRB- cl .. preclosure operator. preclosure operator. Cech closure operator. Cech 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 .. net. net ( mathematics ). Then that closure operator can be shown to satisfy the axioms of a preclosure operator .. preclosure operator. 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 .. idempotent. idempotent. A map f  : -LRB- X , cl -RRB- → -LRB- Y , cl -RRB- between two pretopological spaces is continuous '' ' if it satisfies for all subsets A of X  :. f -LRB- cl -LRB- A -RRB- -RRB- ⊆ cl ' -LRB- f -LRB- A '' -RRB- -RRB- .