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 xα converges to a point x in X if xα 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)) .
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- 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.
- Recombination Spaces, Metrics, and Pretopologies B.M.R. Stadler, P.F. Stadler, M. Shpak., and G.P. Wagner. (See in particular Appendix A.)