In this paper, we associate a topology to G, called graphic topology of G and we show that it is an Alexandroff topology, i.e. a topology in which intersec- tion of. Alexandroff spaces, preorders, and partial orders. 4. 3. Continuous A-space, then the closed subsets of X give it a new A-space topology. We write. Xop for X. trate on the definition of the T0-Alexandroff space and some of its topological . the Scott topology and the Alexandroff topology on finite sets and in general.
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Alexandrov topology – Wikipedia
They should not be confused with the more geometrical Alexandrov spaces introduced by the Russian mathematician Aleksandr Danilovich Aleksandrov. A useful discussion of the abstract relation between posets and Alexandroff locales is in section 4.
Topological spaces satisfying the above equivalent characterizations are called finitely generated spaces or Alexandrov-discrete spaces and their topology T is toplogy an Alexandrov topology.
Alexandrov-discrete spaces are also called finitely generated spaces since their topology is uniquely determined by the family of all finite subspaces. This topology may be strictly coarser, but they are the same if the order is linear. Properties of topological spaces Order theory Closure operators.
The class of modal algebras that we obtain in the case of a preordered set is the class of interior algebras —the algebraic abstractions of topological spaces.
Spaces with this topology, called Alexandroff spaces and named after Paul Alexandroff Pavel Aleksandrovshould not be confused with Alexandrov spaces which arise in differential geometry and are named after Alexander Alexandrov.
Every finite topological space is an Alexandroff space. The latter construction is itself a special case of a more general construction of a complex algebra from a relational structure i.
A function between preorders is order-preserving if and topolog if it is a continuous map with respect to alexandrff specialisation topology. Remark By the definition of the 2-category Locale see therethis means that AlexPoset AlexPoset consists of those morphisms which have right adjoints in Locale. Now, it is clear that Alexandrov topology is at least as big as the upper topology as every principle upper set is indeed an upper set, while the converse need topilogy hold.
Given a monotone function. Email Required, but never shown. The problem is that your definition of the upper topology is wrong: Every Alexandroff space is obtained by equipping its specialization order with the Alexandroff topology.
An Alexandroff topology on graphs
Retrieved from ” https: This is similar to the Scott topologywhich is however coarser. Given a preordered set Xthe interior operator and closure operator of T X are given by:. Proposition A function between preorders is order-preserving if and only if it is a continuous map with respect to the specialisation topology.
Views Read Edit View history. 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.
Proposition The category of Alexandroff locales is equivalent to that of completely distributive algebraic lattice s. Arenas independently proposed this name for the general version of these topologies. Alexandrov topology Ask Question.
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. This page was last edited on 6 Mayat Since the upper topology will have all the principle upper sets as open sets, wouldn’t then the arbitrary union of these open sets will end up generating the whole Alexandrov topology, as every upper set is a union of some set of principle upper sets?
Proposition Every finite topological space is an Alexandroff space.
Notice however that in the case of topologies other than the Alexandrov topology, we can have a map between two topological spaces that is not continuous but which is nevertheless still a monotone function between the corresponding preordered topoloogy.
Sign up using Facebook. Home Questions Tags Users Unanswered. This defines a topology on P Pcalled the specialization topology or Alexandroff topology.
In Michael C. Alexandrov spaces were first introduced in by P. Last revised on April 24, at