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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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 topologies have numerous characterizations. CS1 German-language sources de. Home Questions Tags Users Unanswered. In topologyan Alexandrov topology is a topology in which the intersection of any family of open sets is open.

Thus a map between two preordered sets is monotone if and only alexandrotf it is a continuous map between the corresponding Alexandrov-discrete spaces. Let Top denote the category of topological spaces and continuous maps ; and let Pro denote the category of preordered sets and monotone functions.

This page was last edited on 6 Mayat The category of Alexandroff locales is equivalent to that of completely distributive algebraic lattice s. Last revised on April 24, at 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.

Alexandrov topology – Wikipedia

Hopology 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.


Definition An Alexandroff space is a topological space for which arbitrary as opposed to just finite intersections of open subsets are still open. Alexandrov-discrete spaces can thus be viewed as a generalization of finite topological spaces.

I have also found another definition of the upper topology: This means that given a topological space Xthe identity map. Note that the upper sets are non only a base, they form the whole topology. 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. Proposition Every finite topological space is an Alexandroff space.

Alexandrov-discrete spaces are named after the Russian topologist Pavel Alexandrov.

Thus, Alexandrov topologies on X are in one-to-one correspondence with preorders on X. 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.

I am obviously missing something here. 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. Every finite topological space is an Alexandroff space.

The specialisation topology

Proposition Every Alexandroff space is obtained by equipping its specialization order with the Alexandroff topology. This site is running on Instiki 0.

This defines a topology on P Pcalled the specialization topology or Alexandroff topology. By using this site, you agree to the Terms of Use and Privacy Policy. 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. Declare a subset A A of P P to be an open subset if it is upwards-closed.


Every Alexandroff space is obtained by equipping its specialization order with the Alexandroff topology. Retrieved from ” https: Scott akexandroff 38 A discussion of abelian sheaf cohomology on Alexandroff spaces is in. A set together with an Alexandrov topology is known as an Alexandrov-discrete space or finitely generated space.

Under your definitions, both topologies are the same.

Post as a guest Name. Naturman observed that these spaces were the Alexandrov-discrete spaces and extended the result to a category theoretic duality between the category of Alexandrov-discrete spaces and open continuous maps, and the category of preorders and bounded monotone maps, providing the preorder characterizations as well as the interior and closure algebraic characterizations. 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.

Interior Algebras and Topology. Then W g is a monotone function. To see this consider a non-Alexandrov-discrete space X and consider the identity map i: By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Let Alx denote the full subcategory of Top consisting of the Alexandrov-discrete spaces. The specialisation topologyalso called the Alexandroff topologyis a natural structure of a topological space induced on the underlying set of a preordered set.

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