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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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.

## An Alexandroff topology on graphs

Steiner demonstrated that the duality is a contravariant lattice isomorphism preserving arbitrary meets and joins as well as complementation. Proposition A function between preorders is order-preserving if and only if it is a continuous map with respect to the specialisation topology. The class of modal algebras that we obtain in the case of a preordered set is the class of interior algebras â€”the algebraic abstractions of topological spaces.

Grzegorczyk observed that this extended to a duality between what he referred to as totally distributive spaces and preorders. See toplogy history of this page for a list of all contributions to it.

This defines a topology on P Pcalled the specialization topology or Alexandroff topology.

They should not be confused with the more geometrical Alexandrov spaces introduced by the Russian mathematician Aleksandr Danilovich Aleksandrov. Considering the interior operator and closure operator to be modal operators on the power set Boolean algebra of Xthis construction is a special case alfxandroff the construction of a modal algebra from a modal frame alexqndroff. Let P P tlpology a preordered set. Then the following are equivalent:. This page was last edited on 6 Mayat 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 sets.

By the definition of the 2-category Locale see therethis means that AlexPoset AlexPoset consists of those morphisms which have right adjoints in Locale.

### Alexandrov topology – Wikipedia

Let Alx denote the full subcategory of Top consisting of the Alxandroff spaces. Definition An Alexandroff space is a topological alexanroff for which arbitrary as opposed to just finite intersections of open subsets are still open. The problem is that your definition of the toploogy topology is wrong: And the principal upper sets are only only a subbase, they from a base.

Proposition The functor Alex: This is similar to the Scott topologywhich is however coarser. CS1 German-language sources de.

Mathematics Stack Exchange works best with JavaScript enabled. Alexandrov spaces were first introduced in by P. By clicking “Post Your Answer”, you acknowledge that you have read our updated terms of serviceprivacy policy and cookie policyand that your continued use of the website is subject to these policies.

Properties of topological spaces Order theory Closure operators. Or, upper topology is simply presented with upper sets and their intersections, and nothing more? Proposition Every Alexandroff space is obtained by equipping alexandrofff specialization order with the Alexandroff topology. 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. Sign up or log in Sign up using Google. In Michael C. Retrieved from ” https: With the advancement of categorical topology in the s, Alexandrov spaces were rediscovered when the concept of finite generation was applied to general topology and the name finitely generated spaces was adopted for them. To see this consider a non-Alexandrov-discrete space X and consider the identity topologj i: Sign up using Facebook. Stone spaces 1st paperback ed. Thus a map between two preordered sets is monotone if and only if it is a continuous map between the corresponding Alexandrov-discrete spaces.

Definition Let P P be a preordered set. Every finite topological space is an Alexandroff space. 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.

Johnstone referred to such topo,ogy as Alexandrov topologies.

A set together with an Alexandrov topology is known as an Alexandrov-discrete space or finitely generated space. Alexandrov-discrete spaces are named after the Russian topologist Pavel Alexandrov.

### specialization topology in nLab

This site is running on Instiki 0. Given a monotone function. Proposition The category of Alexandroff locales is equivalent to that of completely distributive algebraic lattice s.

Alexandrov topology Ask Question. But I have the following confusion. By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Sign up using Email and Password. Thus, Alexandrov topologies on X are in one-to-one correspondence with preorders on X. Home Questions Tags Users Unanswered.

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. Alexandrov-discrete spaces can thus be viewed as a generalization of finite topological spaces.