About: Alexandrov topology

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In topology, an Alexandrov topology is a topology in which the intersection of any family of open sets is open. It is an axiom of topology that the intersection of any finite family of open sets is open; in Alexandrov topologies the finite restriction is dropped. A set together with an Alexandrov topology is known as an Alexandrov-discrete space or finitely generated space. Due to the fact that inverse images commute with arbitrary unions and intersections, the property of being an Alexandrov-discrete space is preserved under quotients.

Attributes	Values
rdf:type	yago:WikicatClosureOperators yago:WikicatTopologicalSpaces yago:Abstraction100002137 yago:Attribute100024264 yago:Function113783816 yago:MathematicalRelation113783581 yago:MathematicalSpace108001685 yago:Operator113786413 yago:Possession100032613 yago:Property113244109 yago:Relation100031921 yago:Set107999699 yago:Space100028651 yago:WikicatPropertiesOfTopologicalSpaces
rdfs:label	Alexandrov topology (en) Topología de Aleksándrov (es) Topologie d'Alexandroff (fr) Przestrzeń Aleksandrowa (pl) Topologia de Alexandrov (pt)
rdfs:comment	En mathématiques, une topologie d'Alexandroff est une topologie pour laquelle l'intersection d'une famille quelconque d'ouverts est un ouvert (et pas seulement l'intersection d'une famille finie d'ouverts). Cette notion a été introduite en 1937 par Pavel Alexandroff. Un espace topologique vérifie cette propriété si et seulement si sa topologie est cohérente avec ses sous- (en), c'est pourquoi un tel espace est aussi appelé espace finiment engendré. (fr) Przestrzeń Aleksandrowa – przestrzeń topologiczna, dla której część wspólna dowolnej rodziny jej podzbiorów otwartych jest zbiorem otwartym. Przestrzenie Aleksandrowa zostały zdefiniowane przez Pawła Aleksandrowa w roku 1937 pod nazwą „przestrzenie dyskretne”. (pl) In topology, an Alexandrov topology is a topology in which the intersection of any family of open sets is open. It is an axiom of topology that the intersection of any finite family of open sets is open; in Alexandrov topologies the finite restriction is dropped. A set together with an Alexandrov topology is known as an Alexandrov-discrete space or finitely generated space. Due to the fact that inverse images commute with arbitrary unions and intersections, the property of being an Alexandrov-discrete space is preserved under quotients. (en) En matemática, a cualquier preorden se le puede dar la estructura de un espacio topológico, declarando abierta cualquier sección final (conjunto superior). Se puede demostrar que cualquier topología «fina» viene de esa debido al (pre)orden de especialización y, entre tales espacios, una función es continua si y solamente si es . Esto contesta a una buena pregunta: si toda intersección (no sólo las intersecciones finitas) de conjuntos abiertos es abierta. Respuesta: esta topología es de Alexandrov (también escrito Alexandroff), en honor a Pável Aleksándrov, quien fue el primero en estudiarlas. (es) Em topologia, um Espaço de Alexandrov é um espaço topológico no qual a intersecção de qualquer família de abertos é aberta. Há um axioma da topologia que diz que qualquer intersecção finita de abertos é um aberto, mas em um espaço de Alexandrov, a restrição de ser finito é descartada. (pt)
dct:subject	Order theory Properties of topological spaces Closure operators
Wikipage page ID	542198 (xsd:integer)
Wikipage revision ID	1107485657 (xsd:integer)
Link from a Wikipage to another Wikipage	Power set Preorder Modal logic Coreflective subcategory Binary relation Denotational semantics Interior algebra Order theory General topology One-to-one correspondence Closure operator Alexandrov space Properties of topological spaces Full subcategory Andrzej Grzegorczyk Locally compact Complete lattice Computer science Øystein Ore Kripke semantics Adjoint functors Category of sets Topology Domain theory Lattice (order) Aleksandr Danilovich Aleksandrov Partially ordered set Kolmogorov space Continuity (topology) Quotient space (topology) Intersection (set theory) Closure operators Coherent topology Modal algebra Modal frame Upper set Discrete space Boolean algebra (structure) Specialization preorder Identity function Open set Order complex Category of topological spaces Set (mathematics) Map (mathematics) Complex algebra (set theory) Image (mathematics) Finite topological space Finitely generated object Lattice theory Topological space P-space Lower set Preordered set P. S. Alexandrov P. T. Johnstone Finest topology Categorical topology Covariance and contravariance of functors Relational structure Continuous map (topology) Interior operator Union (mathematics) Local base Locally path connected Monotone function Neighbourhood (topology) Neighbourhood filter Weak homotopy equivalence dbr:Concrete_functor dbr:Final_hull

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