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.

Let "X" be a set. A neighborhood system for a pretopology on X is a collection of filters "N"("x"), one for each element 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.

External links

* [http://www.tbi.univie.ac.at/papers/Abstracts/01-02-011.pdf Recombination Spaces, Metrics, and Pretopologies] B.M.R. Stadler, P.F. Stadler, M. Shpak., and G.P. Wagner. (See in particular Appendix A.)