# Pretopological space

In general topology, a **pretopological space** is a generalization of the concept of topological space. A pretopological space can be defined as 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*_{α} converges to a point *x* in *X* if *x*_{α} is eventually in every neighborhood of *x*.

A pretopological space can also be defined as (*X*, *cl* ), a set *X* with a preclosure operator (Čech closure operator) *cl*. 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 *f* : (*X*, *cl* ) → (*Y*, *cl'* ) between two pretopological spaces is **continuous** if it satisfies for all subsets *A* of *X*:

*f*(*cl*(*A*)) ⊆*cl'*(*f*(*A*)) .

## References

- E. Čech,
*Topological Spaces*, John Wiley and Sons, 1966. - D. Dikranjan and W. Tholen,
*Categorical Structure of Closure Operators*, Kluwer Academic Publishers, 1995. - S. MacLane, I. Moerdijk,
*Sheaves in Geometry and Logic*, Springer Verlag, 1992.

## External links

- Recombination Spaces, Metrics, and Pretopologies B.M.R. Stadler, P.F. Stadler, M. Shpak., and G.P. Wagner. (See in particular Appendix A.)