In topology, an Alexandrov topology is a topology in which the intersection of every family of open sets is open. It is an axiom of topology that the intersection of every 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.
Alexandrov topologies are uniquely determined by their specialization preorders. Indeed, given any preorder ≤ on a set X, there is a unique Alexandrov topology on X for which the specialization preorder is ≤. The open sets are just the upper sets with respect to ≤. Thus, Alexandrov topologies on X are in one-to-one correspondence with preorders on X.
Alexandrov-discrete spaces are also called finitely generated spaces because their topology is uniquely determined by the family of all finite subspaces. Alexandrov-discrete spaces can thus be viewed as a generalization of finite topological spaces.
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.
Alexandrov-discrete spaces are named after the Russian topologist Pavel Alexandrov. They should not be confused with the more geometrical Alexandrov spaces introduced by the Russian mathematician Aleksandr Danilovich Aleksandrov.
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In topology, an Alexandrovtopology is a topology in which the intersection of every family of open sets is open. It is an axiom of topology that the...
to set theory and topology. In topology, the Alexandroff compactification and the Alexandrovtopology are named after him. Alexandrov attended Moscow State...
required to be open. This topology goes back to Pavel Alexandrov. Nowadays, the correct mathematical term for the Alexandrovtopology on spacetime (which goes...
general topology, a field of mathematics. Alexandrovtopology Cantor space Co-kappa topology Cocountable topology Cofinite topology Compact-open topology Compactification...
corresponding Alexandrovtopology, an alternative representation of the interior algebra as a field of sets is obtained where the topology is the Alexandrov bico-reflection...
obtained by Li Chen and Yongwu Rong. A 2D grid cell topology already appeared in the Alexandrov–Hopf book Topologie I (1935). Hauptvermutung Topological...
consistent topology is the Scott topology, which is coarser than the Alexandrovtopology. A third important topology in this spirit is the Lawson topology. There...
In mathematics, the poset topology associated to a poset (S, ≤) is the Alexandrovtopology (open sets are upper sets) on the poset of finite chains of...
(pre)order Sober space Spectral space Alexandrovtopology Upper topology Scott topology Scott continuity Lawson topology Polish Space Cantor space Inductive...
measurable. Alexandrovtopology Lexicographic order topology on the unit square Order topology Lawson topology Poset topology Upper topology Scott topology Scott...
be considered as a link to classic combinatorial topology, appeared in the book of Pavel Alexandrov and Heinz Hopf, Topologie I (1935). Rosenfeld et al...
be given a topology, the Alexandrovtopology; and indeed, every preorder on a set is in one-to-one correspondence with an Alexandrovtopology on that set...
closure Topology & Orders Alexandrovtopology & Specialization preorder Ordered topological vector space Normal cone Order topology Order topology Topological...
Order topology of a total order (open interval topology) Alexandrovtopology Upper topology Scott topology Scott continuity Lawson topology Finer topology...
particular point topology that doesn't contain the particular point, has the discrete topology. Alexandrovtopology Excluded point topology Finite topological...
from F∗ onto ±1. Giving ±1 the discrete topology and ±1F the product topology induces the subspace topology on XF. The Harrison sets H ( a ) = { P ∈...
mathematics, an order topology is a specific topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real...
Alexandrovtopology we obtain an alternative representation of the interior algebra as a topological field of sets. (The topology of this "Alexandrov...
define a topology on any ordered set, the order topology. When more than one order is being used on a set one talks about the order topology induced by...
τ). The topology defined in this way is called the Alexandrovtopology determined by ≤. The equivalence between preorders and finite topologies can be...
Galois connection. Alexandrovtopology. For a preordered set P, any upper set O is Alexandrov-open. Inversely, a topology is Alexandrov if any intersection...
doi:10.1017/9781316543870. ISBN 9781107146723. S2CID 201542298. Alexandrovtopology Superintuitionistic (aka intermediate) logics List of Boolean algebra...
Hausdorff so it is not locally compact in sense (5). Every space with an Alexandrovtopology is locally compact in senses (1) and (3). Every locally compact preregular...
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