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.

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 <math>X</math> be a set. A neighborhood system for a pretopology on <math>X</math> is a collection of filters <math>N(x)</math>, one for each element <math>x</math> of <math>X</math>, such that every set in <math>N(x)</math> contains <math>x</math> as a member. Each element of <math>N(x)</math> is called a neighborhood of <math>x.</math> A pretopological space is a set equipped with such a neighborhood system.

A net <math>x_{\alpha}</math> converges to a point <math>x</math> in <math>X</math> if <math>x_{\alpha}</math> is eventually in every neighborhood of <math>x.</math>

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

See also

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.
  • 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 .