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