Pretopological space information

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