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

Mathematics Stack Exchange works best with JavaScript enabled. Proposition Every finite topological space is an Alexandroff space. Then W g is a monotone function. Notice however that in the case alexandtoff topologies other than the Alexandrov topology, we can have a tooology between two topological spaces that is not continuous but which is nevertheless still a monotone function between the corresponding preordered sets.

This is similar to the Scott topologywhich is however coarser.

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. Thus, Alexandrov topologies on X are in one-to-one correspondence with preorders on X. Alexandrov spaces were also rediscovered around the alrxandroff time in the context of topologies resulting from denotational semantics and domain theory in computer science.

Under your definitions, both topologies are the same. Sign up or log in Sign up using Google. Let Top denote the category of topological spaces and continuous maps ; and let Pro denote the category of preordered sets and monotone functions. Arenas independently proposed this name for the general version of these topologies. They are alexnadroff the same for every linear order. The category of Alexandroff locales is equivalent to that of completely distributive algebraic lattice s.

### specialization topology in nLab

And the principal upper sets are only only a subbase, they from a base. Given a preordered set Xthe interior operator and closure operator of T X are given by:.

Thus a map between two preordered sets is monotone if and only if it is a continuous map between the corresponding Alexandrov-discrete spaces. By using our site, you acknowledge that you have alexsndroff and understand our Cookie PolicyPrivacy Policyand our Terms of Service. A set together with an Alexandrov topology is known as an Alexandrov-discrete space or finitely generated space.

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. Post as a guest Name. They should not be confused with the more geometrical Alexandrov spaces introduced by the Russian mathematician Aleksandr Danilovich Aleksandrov. This page was last edited on 6 Mayat This defines a topology on P Pcalled the specialization topology or Alexandroff topology.

Alexandrov topologies have numerous characterizations.

CS1 German-language sources de. Conversely a map between two Alexandrov-discrete spaces is continuous if and only if it is a monotone function between the corresponding preordered sets. Sign up using Facebook. Alexandrov-discrete spaces are also called finitely generated spaces since their topology is uniquely determined by the family of all finite subspaces.

Note that the upper sets are non only a base, they form the whole topology. Proposition Every Alexandroff space is obtained by equipping its specialization order with the Alexandroff topology.

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. In Michael C. Alexandrov spaces were first introduced in by P. By using this site, you agree to the Terms of Use and Privacy Policy.

## An Alexandroff topology on graphs

Views Read Edit View history. Definition An Alexandroff space is a topological space for which arbitrary as opposed to just finite intersections of open subsets are still open. This topology may be strictly coarser, but they are the same if the order is linear. Alexandrov under the name discrete spaceswhere he provided the characterizations in terms of sets and neighbourhoods.

Proposition A function between preorders is order-preserving if and only if it is a continuous map with respect to the specialisation topology. Let P P be a preordered set. A function between preorders is order-preserving if and only if it is a continuous map with respect to the specialisation topology. Let Alx denote the full subcategory of Top consisting of the Alexandrov-discrete spaces. This means that given a topological space Xthe identity map.

Home Questions Tags Users Unanswered. Let Set denote the category of sets and maps. Retrieved from ” https: The corresponding closed sets are the lower sets:. 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.

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

Transactions of the American Mathematical Society.