Pretopological space

{{Short description|Generalized topological 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 $x\_\{\backslash alpha\}$ converges to a point $x$ in $X$ if $x\_\{\backslash alpha\}$ is eventually in every neighborhood of $x.$

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