Pretopological space

Short description: Generalized topological 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.

Let [math]\displaystyle{ X }[/math] be a set. A neighborhood system for a pretopology on [math]\displaystyle{ X }[/math] is a collection of filters [math]\displaystyle{ N(x), }[/math] one for each element [math]\displaystyle{ x }[/math] of [math]\displaystyle{ X }[/math] such that every set in [math]\displaystyle{ N(x) }[/math] contains [math]\displaystyle{ x }[/math] as a member. Each element of [math]\displaystyle{ N(x) }[/math] is called a neighborhood of [math]\displaystyle{ x. }[/math] A pretopological space is then a set equipped with such a neighborhood system.

A net [math]\displaystyle{ x_{\alpha} }[/math] converges to a point [math]\displaystyle{ x }[/math] in [math]\displaystyle{ X }[/math] if [math]\displaystyle{ x_{\alpha} }[/math] is eventually in every neighborhood of [math]\displaystyle{ x. }[/math]

A pretopological space can also be defined as [math]\displaystyle{ (X, \operatorname{cl}), }[/math] a set [math]\displaystyle{ X }[/math] with a preclosure operator (Čech closure operator) [math]\displaystyle{ \operatorname{cl}. }[/math] The two definitions can be shown to be equivalent as follows: define the closure of a set [math]\displaystyle{ S }[/math] in [math]\displaystyle{ X }[/math] to be the set of all points [math]\displaystyle{ x }[/math] such that some net that converges to [math]\displaystyle{ x }[/math] is eventually in [math]\displaystyle{ S. }[/math] Then that closure operator can be shown to satisfy the axioms of a preclosure operator. Conversely, let a set [math]\displaystyle{ S }[/math] be a neighborhood of [math]\displaystyle{ x }[/math] if [math]\displaystyle{ x }[/math] is not in the closure of the complement of [math]\displaystyle{ S. }[/math] 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 [math]\displaystyle{ f : (X, \operatorname{cl}) \to (Y, \operatorname{cl}') }[/math] between two pretopological spaces is continuous if it satisfies for all subsets [math]\displaystyle{ A \subseteq X, }[/math] [math]\displaystyle{ f(\operatorname{cl}(A)) \subseteq \operatorname{cl}'(f(A)). }[/math]

See also

• Kuratowski closure axioms
• Cauchy space – Concept in general topology and analysis
• Convergence space – Generalization of the notion of convergence that is found in general topology
• Proximity space – Structure describing a notion of "nearness" between subsets

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.)
• Closed sets and closures in Pretopology M. Dalud-Vincent, M. Brissaud, and M Lamure. 2009 .