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

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

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

• 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

• 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.

• 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 .

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