Pretopology

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 $X$ 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 $x_$ converges to a point $x$ in $X$ if $x_$ is eventually in every neighborhood of $x.$

A pretopological space can also be defined as $(X, \operatorname),$ a set $X$ with a preclosure operator (ÄŒech closure operator) $\operatorname.$ 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 $f : (X, \operatorname) \to (Y, \operatorname')$ between two pretopological spaces is continuous if it satisfies for all subsets $A \subseteq X,$ $f(\operatorname(A)) \subseteq \operatorname'(f(A)).$

See also

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References

* E. ÄŒech, ''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 .

{{Topology

General topology