# ALEXANDROFF TOPOLOGY PDF

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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Proposition The category of Alexandroff locales is equivalent to that of completely distributive algebraic lattice s. 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 allexandroff principle upper sets?

A discussion of abelian sheaf cohomology on Alexandroff spaces is in. Let Top denote the category of topological spaces and continuous maps ; and let Pro denote the category of alexandrodf sets and monotone functions.

### order theory – Upper topology vs. Alexandrov topology – Mathematics Stack Exchange

Let Alx denote the full subcategory of Top consisting of the Alexandrov-discrete spaces. This page was last edited on 6 Mayat CS1 German-language sources de.

To see this consider a non-Alexandrov-discrete space X and consider alexandrroff identity tppology i: 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. Alexandrov-discrete spaces are also called finitely generated spaces since their topology is uniquely determined by the family of all finite subspaces.

Johnstone referred to such topologies as Alexandrov topologies. Retrieved from ” https: By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Alexandrov topologies are uniquely determined by their specialization preorders. The latter construction is itself a special case of a more general construction of a complex algebra from a relational structure i.

This defines a topology on P Pcalled the specialization topology or Alexandroff topology. Arenas, Alexandroff spacesActa Math. Sign up using Email and Password. A useful discussion of the abstract relation between posets and Alexandroff locales is in section 4.

In Michael C. Grzegorczyk topoology that this extended to a duality between what he referred to as totally distributive spaces and preorders. 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.

In topologyan Alexandrov topology is a topology in which the intersection of any family of open sets is open. With the advancement of categorical topology in the s, Alexandrov spaces were rediscovered when the concept of finite generation was applied fopology general topology and the name finitely generated spaces was adopted for them. 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.

Proposition Every finite topological space is an Alexandroff space. Alexnadroff spaces are named after the Russian alfxandroff Pavel Alexandrov. Every finite topological space is an Alexandroff space.

Thus a map between two preordered sets is monotone if and only if it is a continuous map between the corresponding Alexandrov-discrete spaces.

### An Alexandroff topology on graphs

Conversely a map between two Alexandrov-discrete spaces is continuous if and only if it is a monotone function between the corresponding preordered sets. 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. Considering the interior operator and closure operator alexanrdoff be modal operators on the power set Boolean algebra of Xthis construction is a special case of the construction of a modal algebra from aoexandroff modal frame i.

I have also found another definition of the upper topoloyg Declare a subset A A of P P to be an open subset if it is upwards-closed. Alexandroft as a guest Name. Scott k 38 Every Alexandroff space is obtained by equipping its specialization order with the Alexandroff topology. Given a monotone function. They should not be confused with the more geometrical Alexandrov spaces introduced by the Russian mathematician Aleksandr Danilovich Aleksandrov.

Sign up or log in Sign up using Google. 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.

Alexandrov spaces were first introduced in by P.

## The specialisation topology

See the history of this page for a list of all contributions to it. Alexandrov under the name discrete spaceswhere he provided the characterizations in terms of sets and neighbourhoods.

Proposition A function between preorders tooplogy order-preserving if and only if it is a continuous map with respect to the specialisation topology.

Views Read Edit View history. 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.

Transactions of the American Mathematical Society. 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.

## An Alexandroff topology on graphs

Proposition The functor Alex: Definition Let P P be a preordered set. Let P P be a preordered set. I am obviously missing something here.