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 {\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 then 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 it satisfies for all subsets A ⊆ X , {\displaystyle A\subseteq X,} f ( cl ⁡ ( A ) ) ⊆ cl ′ ⁡ ( f ( A ) ) . {\displaystyle f(\operatorname {cl} (A))\subseteq \operatorname {cl} '(f(A)).}