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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Declare a subset A A of P P to be an open subset if it is upwards-closed. Alexandrov spaces were first introduced in by P.

And the principal upper sets are only only a subbase, they from a base. In Michael C. Sign up using Facebook. Post Your Answer Discard 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.

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. They should not be confused with the more geometrical Alexandrov spaces introduced by the Russian mathematician Aleksandr Danilovich Aleksandrov. Every finite topological space is an Alexandroff space.

## The specialisation topology

Now, it is clear that Alexandrov topology is at alexxndroff as big as the upper topology as every principle upper set is indeed an upper set, while the converse need not hold. 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.

I have also found another definition of the upper topology: By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

The latter construction is itself a special case of a more general construction of a complex algebra from a relational structure i. 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.

A set together with allexandroff Alexandrov topology is known as an Alexandrov-discrete space or finitely generated space. Johnstone referred to such topologies as Alexandrov topologies.

Proposition Every Alexandroff space is obtained by equipping its specialization order with the Alexandroff topology. Post as a guest Name.

## An Alexandroff topology on graphs

Alexandrov topology Ask Question. Sign up or log in Sign up using Google. Every Alexandroff space is obtained by equipping its specialization order with the Alexandroff topology. Let Alx denote the full subcategory of Top consisting of the Alexandrov-discrete spaces. Then the following are equivalent:. Alexandrov-discrete spaces are named after the Russian topologist Pavel Alexandrov. Interior Algebras and Topology.

Conversely a map between two Alexandrov-discrete spaces is continuous if and only if it is a monotone function between the corresponding preordered sets. A function between preorders is order-preserving if and only if it is a continuous map with respect to the specialisation topology.

A discussion of abelian sheaf cohomology on Alexandroff spaces is in. Alexandrov-discrete spaces are also called finitely generated spaces since their topology is uniquely determined by the family of all finite subspaces.

The specialisation topologyalso called the Alexandroff topologyis a natural structure of a topological space induced on the underlying set of a preordered set. Mathematics Stack Exchange works best with JavaScript enabled. 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.

### Alexandrov topology – Wikipedia

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 alexansroff website is subject to these policies. The category of Alexandroff locales is equivalent to that of completely distributive algebraic lattice s.

It was also alexandrovf 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. Definition Let P P be a preordered set. Views Read Edit View history.