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