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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The specialisation topologyalso called the Alexandroff topologyis a natural structure of a topological space induced on the underlying set of a preordered set.

This is similar to the Scott topologywhich is however coarser. 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.

Let P P be a preordered set. Declare a subset A A of P P to be an open subset if it is upwards-closed. This defines a topology on P Pcalled the specialization topology or Alexandroff topology.

### Alexandrov topology – Wikipedia

Every finite topological space is an Alexandroff space. A function between preorders is order-preserving if and only if it is a continuous map with respect to the specialisation topology.

An Alexandroff space is a topological space for which arbitrary as opposed to just finite intersections of open subsets are still open. Every Alexandroff space is obtained topolohy equipping its specialization order with the Alexandroff topology. By the definition of the 2-category Locale see therethis means that AlexPoset AlexPoset consists of those morphisms which have right adjoints alexzndroff Locale.

The category of Alexandroff locales is equivalent to that of completely distributive slexandroff lattice s. Arenas, Alexandroff spacesActa Math.

A useful discussion of the abstract relation between posets and Alexandroff locales is in section 4.

A discussion of abelian sheaf cohomology on Alexandroff spaces is in. Alexandroft revised on April 24, at See the history of this page for a list of all contributions to it.

Definition Let P P be a preordered set.

## Alexandrov topology

Proposition Every finite topological space is an Alexandroff space. Proposition A function between preorders is order-preserving if and only if it is a continuous map with respect to the specialisation topology.

Definition An Alexandroff space is a topological space for which arbitrary as opposed to just finite intersections of open subsets are still open.

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

### An Alexandroff topology on graphs

Remark By the definition qlexandroff the 2-category Locale see therethis means that AlexPoset AlexPoset consists of those morphisms which have right adjoints in Locale. Proposition The functor Alex: Proposition The category of Alexandroff locales is equivalent to that of completely distributive algebraic lattice s.

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