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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The problem is that your definition of the upper topology is wrong: Alexandrov topologies topolkgy numerous characterizations. Grzegorczyk observed that this extended to a duality between what he referred to as totally distributive spaces and preorders. This page was last edited on 6 Mayat Definition An Alexandroff space is a topological space for which arbitrary as opposed to just finite intersections of open subsets are still open.

## An Alexandroff topology on graphs

This means that given a topological space Xthe identity map. Let P P be a preordered set. Then W g is a monotone function.

Alexandrov-discrete spaces are also called finitely generated spaces since their topology is uniquely determined by the family of all finite subspaces. A set together with an Alexandrov topology is known as an Alexandrov-discrete space or topoloyg generated space.

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.

The latter construction is itself a special case of a more general construction of a complex algebra from a relational structure i. Last revised on April 24, at The specialisation topologyalso called the Alexandroff topologyis a natural structure of a topological space induced on the underlying set topologt a preordered set. Proposition A function between preorders is order-preserving if and only if it is a continuous map with respect to the specialisation topology.

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. Post as a guest Name.

Home Questions Tags Users Unanswered. Stone spaces 1st paperback ed. 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.

The category of Alexandroff locales is equivalent to that of completely distributive algebraic lattice s. Sign up using Facebook.

### specialization topology in nLab

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. Alexandrov spaces were first introduced in by P. Email Required, but never shown. Let Slexandroff denote the full subcategory of Top consisting of the Alexandrov-discrete spaces. Conversely a map between two Alexandrov-discrete spaces is continuous if and only if it is a monotone function between the corresponding preordered sets.

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

Thus a map between two preordered sets is topolovy if and only if it is a continuous map between the corresponding Alexandrov-discrete spaces. Then T f is a continuous map. Note that the upper sets are non only a base, they form the whole topology.

### An Alexandroff topology on graphs

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Alexandrov topologies are uniquely determined by their specialization preorders. Proposition Every finite topological space is an Alexandroff space. I am obviously missing something here. By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Scott k 38 This site is running on Instiki 0. 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?

A systematic investigation of these spaces from the point of view of general topology which had been neglected since the original paper by Alexandrov, was taken up by F. 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. In topologyan Alexandrov topology is a topology in which the intersection of any family of open sets is open.

CS1 German-language sources de. In Michael C.

Interior Algebras and Topology. This is similar to the Scott topologywhich is however coarser. Mathematics Stack Exchange works best with JavaScript enabled. But I have the following confusion. Naturman observed that these spaces were the Alexandrov-discrete spaces and fopology the result to a category theoretic duality between the category of Alexandrov-discrete spaces and open alexanvroff maps, and the category of preorders and bounded monotone maps, providing the preorder characterizations as well as the interior and closure algebraic characterizations.