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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 , {\displaystyle N,} one for each element x {\displaystyle x} of X {\displaystyle X} such that every set in N {\displaystyle N} contains x {\displaystyle x} as a member. Each element of N {\displaystyle N} 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 , {\displaystyle ,} a set X {\displaystyle X} with a preclosure 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.