Pretopological 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 $X$ 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 $x_{\alpha }$ converges to a point $x$ in $X$ if $x_{\alpha }$ is eventually in every neighborhood of $x.$

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

$$f(\operatorname {cl} (A))\subseteq \operatorname {cl} '(f(A)).$$

See also

• Kuratowski closure axioms – mathematical conceptPages displaying wikidata descriptions as a fallback
• 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 .