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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 be a set. A neighborhood system for a pretopology on is a collection of filters one for each element of such that every set in contains as a member. Each element of is called a neighborhood of A pretopological space is then a set equipped with such a neighborhood system. A net converges to a point in if is eventually in every neighborhood of A pretopological space can also be defined as a set with a preclosure operator (Čech closure operator) The two definitions can be shown to be equivalent as follows: define the closure of a set in to be the set of all points such that some net that converges to is eventually in Then that closure operator can be shown to satisfy the axioms of a preclosure operator. Conversely, let a set be a neighborhood of if is not in the closure of the complement of 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 between two pretopological spaces is continuous if it satisfies for all subsets (en)
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- 3131 (xsd:nonNegativeInteger)
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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. A net converges to a point in if is eventually in every neighborhood of A pretopological space is a topological space when its closure operator is idempotent. A map between two pretopological spaces is continuous if it satisfies for all subsets (en)
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- Pretopological space (en)
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