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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Let Alx denote the full subcategory of Top consisting of the Alexandrov-discrete spaces. Steiner demonstrated that the duality is a contravariant lattice isomorphism preserving arbitrary meets and joins as well as topklogy. It was also a well known result in the field of modal logic that a duality exists between finite topological spaces and preorders on finite sets the finite modal frames for the modal logic S4.
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.
Stone spaces 1st paperback ed. Transactions of the American Alexsndroff Society. By the definition of the 2-category Locale see therethis means that AlexPoset AlexPoset consists of those morphisms which have right adjoints in Locale.
Let Set denote the category of sets and maps. A set together with an Alexandrov topology is known as an Alexandrov-discrete space or finitely generated space.
Alexandrov-discrete spaces can thus be viewed as a generalization of finite topological spaces. This page was last edited on 6 Mayat Proposition A function between preorders is order-preserving if and only if it is a continuous map with respect to the specialisation topology.
Topological spaces satisfying the above equivalent characterizations are called finitely generated spaces or Alexandrov-discrete spaces and their topology T is called an Alexandrov topology. Declare a subset A A of P P to be an open subset if it is upwards-closed.
This is similar to the Scott topologywhich is however coarser. They should not be confused with the more geometrical Alexandrov spaces introduced by the Russian mathematician Aleksandr Danilovich Aleksandrov. Alexandroft site is running tlpology Instiki 0. Alexandrofv 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.
specialization topology in nLab
Alexandrov-discrete spaces are named after the Russian topologist Pavel Alexandrov. Let P P be a preordered set. Considering the interior operator and closure operator to be modal operators on the power set Boolean algebra of Xthis construction is a special case of the construction of a modal algebra from a modal frame i.
See the history of this page for a list of all contributions to it. Then W g is a monotone function. The corresponding closed sets are the lower sets:. Proposition Every Alexandroff space is obtained by equipping its specialization order with the Alexandroff topology.
Proposition The category of Alexandroff locales is equivalent to that of completely distributive algebraic lattice s. Every Alexandroff space is obtained by equipping its specialization order with the Alexandroff topology. The specialisation topologyalso called the Alexandroff topologyis a natural structure of a topological space induced on the underlying set of a preordered set.
Thus a map between two preordered sets is monotone if and only if it is a continuous map between the corresponding Alexandrov-discrete spaces.
But I have the following confusion. Then T f is a continuous map. I have also found another definition of the upper topology: Alexandrov spaces were also rediscovered around the same time in the context of topologies resulting from denotational semantics and domain theory in computer science.
Grzegorczyk observed that this extended to a duality between what he referred to as totally distributive spaces and preorders.
They are not the same for every linear order. 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 not hold. Johnstone referred to such topologies as Alexandrov topologies. Given a monotone function.
An Alexandroff topology on graphs
Steiner each independently observed a duality between partially ordered sets and spaces which were precisely the T 0 versions of the spaces that Alexandrov had introduced. Proposition Every finite topological space is an Alexandroff space. The problem is that your definition of the upper topology topplogy wrong: This topology may be strictly coarser, but they are the same if the order is linear.
Arenas independently proposed this name for the general version of these topologies.
Note that the upper sets are non only a base, they form the whole topology. From Wikipedia, the free encyclopedia. The category of Alexandroff locales is equivalent to that of completely distributive algebraic lattice s. Notice however that in the case of topologies other than the Alexandrov topology, we can have a map between alexndroff topological spaces that is not continuous but which is nevertheless still a monotone function between the corresponding preordered sets.
Properties of topological spaces Alexandrofc theory Closure operators. 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.
This defines a topology on P Pcalled the specialization topology or Alexandroff topology. Under your definitions, both topologies are the same.