Pretopological space

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.

A neighborhood system for a pretopology on

is a collection of filters

A pretopological space is then a set equipped with such a neighborhood system.

converges to a point

is eventually in every neighborhood of

A pretopological space can also be defined as

with a preclosure operator (Čech closure operator)

The two definitions can be shown to be equivalent as follows: define the closure of a set

to be the set of all points

such that some net that converges to

Then that closure operator can be shown to satisfy the axioms of a preclosure operator.

Conversely, let a set

is not in the closure of the complement of

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.

between two pretopological spaces is continuous if it satisfies for all subsets