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.

Sign up using Email and Password. Alexandrov topologies are uniquely determined by their specialization preorders. Conversely a map between two Alexandrov-discrete spaces is continuous if and only if it is a topoloogy function between the corresponding preordered sets.

Declare a subset A Alexandgoff of P P to be an open subset if it is upwards-closed. Proposition A function between preorders is order-preserving if and only if it is a continuous map with respect to the specialisation topology.

Alexandrov under the name discrete spaceswhere he provided the characterizations in terms of sets and neighbourhoods.

Alexandrov topology Ask Question. In topologyan Alexandrov topology is a topology in which the intersection of any family of open sets is open. Arenas, Alexandroff spacesActa Math. Or, upper topology is simply presented with upper sets and their intersections, and nothing more? The corresponding closed sets are the lower sets:. Let P P be a preordered set. From Wikipedia, the free encyclopedia. The problem is that your definition of the upper topology is alexanddroff Views Read Edit View history.

### An Alexandroff topology on graphs

Sign up or log in Sign up using Google. Scott k 38 Johnstone referred to such topologies as Alexaandroff topologies. To see this consider a non-Alexandrov-discrete space X and consider the identity map i: Every Alexandroff space is obtained by equipping its specialization order with the Alexandroff topology. A set together with an Alexandrov topology is known as an Alexandrov-discrete space or finitely generated space.

## The specialisation topology

Definition Let P P be a preordered set. Note that the upper sets are non only a base, they form the whole topology.

Proposition The functor Alex: They are not the same for every linear order. This defines a topology on Alexandrlff Pcalled the specialization topology or Alexandroff topology.

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.

A discussion of abelian sheaf cohomology on Alexandroff spaces is in.

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

But I have the following confusion. By clicking “Post Your Answer”, you acknowledge that you have read our updated terms of serviceprivacy aledandroff and cookie policyand that your continued use of the website is subject to these policies. Under your definitions, both topologies are the same. 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.

Alexandrov-discrete spaces are named after the Russian topologist Pavel Alexandrov.