In general topology, a pretopological space is a generalization of the concept of a topological space. A pretopological space can be defined in terms of either filters or a preclosure operator. Let X {\displaystyle X} be a set. A neighborhood system for a pretopology on X {\displaystyle X} is a collection of filters N ( x ) {\displaystyle N(x)} , one for each element x {\displaystyle x} of X {\displaystyle X} , such that every set in N ( x ) {\displaystyle N(x)} contains x {\displaystyle x} as a member. Each element of N ( x ) {\displaystyle N(x)} is called a neighborhood of x . {\displaystyle x.} A pretopological space is a set equipped with such a neighborhood system. A net x α {\displaystyle x_{\alpha }} converges to a point x {\displaystyle x} in X {\displaystyle X} if x α {\displaystyle x_{\alpha }} is eventually in every neighborhood of x . {\displaystyle x.}
A pretopological space can also be defined as ( X , cl ) , {\displaystyle (X,\operatorname {cl} ),} a set X {\displaystyle X} with a preclosure operator (Čech closure operator) cl . {\displaystyle \operatorname {cl} .} The two definitions can be shown to be equivalent as follows: define the closure of a set S {\displaystyle S} in X {\displaystyle X} to be the set of all points x {\displaystyle x} such that some net that converges to x {\displaystyle x} is eventually in S {\displaystyle S} . Then that closure operator can be shown to satisfy the axioms of a preclosure operator. Conversely, let a set S {\displaystyle S} be a neighborhood of x {\displaystyle x} if x {\displaystyle x} is not in the closure of the complement of S {\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 f : ( X , cl ) → ( Y , cl ′ ) {\displaystyle f:(X,\operatorname {cl} )\to (Y,\operatorname {cl} ')} between two pretopological spaces is continuous if, for all subsets A ⊆ X {\displaystyle A\subseteq X} , we have f ( cl ( A ) ) ⊆ cl ′ ( f ( A ) ) . {\displaystyle f(\operatorname {cl} (A))\subseteq \operatorname {cl} '(f(A)).}
See also Kuratowski closure axioms – Axioms for defining a topology 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 .
