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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This is similar to the Scott topologywhich is however coarser. The category of Alexandroff locales is equivalent to that of completely distributive algebraic lattice s.

See the history of this page for a list of all contributions to it. By using our site, you acknowledge that you have read qlexandroff understand our Cookie PolicyPrivacy Policyand our Terms of Service. 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-discrete spaces are also called finitely generated spaces since their topology is uniquely determined by the family of all finite subspaces. Proposition Every Alexandroff space is obtained topilogy equipping its specialization order with the Alexandroff topology. They should not be confused with the more geometrical Alexandgoff spaces introduced by the Russian mathematician Aleksandr Danilovich Aleksandrov.

In Michael C. 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 topology – Wikipedia

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Every Alexandroff space is obtained by equipping its specialization order with the Alexandroff topology. 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.

Proposition The functor Alex: Post as a guest Name. Retrieved from ” https: Alexandrov spaces were also rediscovered around the same time in the context of topologies resulting from denotational semantics and domain theory in computer science. 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.

Let Set denote the category of sets and maps.

## An Alexandroff topology on graphs

Given a preordered set Xthe interior operator and closure operator of T X are given by:. Declare a subset A A of P P to be an open subset if it is upwards-closed.

Note that the upper sets are non only a base, they form the alexansroff topology. Stone spaces 1st paperback ed.

Alexandrov topology Ask Question. Definition Let P P be a preordered set. This means that given a topological space Xthe identity map. But I have the following confusion. This topology may be strictly coarser, but they are the same if the order is linear.

The corresponding closed sets are the lower sets:. Mathematics Stack Exchange works best with JavaScript enabled. Steiner demonstrated that the duality is a contravariant lattice alexqndroff preserving arbitrary meets and joins as well as complementation. Grzegorczyk observed that this extended to a duality between what he referred to as totally distributive spaces and preorders.

Arenas independently proposed this name for the general version of these topologies. By clicking “Post Your Answer”, you acknowledge that you have read our updated terms of serviceprivacy policy and cookie alexwndroffand that your continued use of the website is subject to these policies.

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 problem is that your definition of the upper topology is wrong: Interior Algebras and Topology.

The latter construction is itself a special case of a more general construction of a complex algebra from a relational structure i. Transactions of the American Mathematical Society.