The Alexandrov topology is a topology that can be put on any poset.
Let be a poset. Then the open sets of the Alexandrov topology are the upper sets of , i.e. the subsets such that
Think of a poset as consisting of some "finite pieces of information" and of as meaning that has at least as much information as .
Then, each open set of the Alexandrov topology can be seen as one that is closed under information extension: if then also contains all elements that have more information than . If we can observe in a finitistic way (e.g. in finite time), then we can establish the property of being in in a finite way.
Thus, the Alexandrov topology can be seen as capturing a logic of finite observation.
If the elements of are to be seen as infinitary observations, then the Scott topology is more appropriate instead.
These intuitions come from [Vickers 1989, Chapter 3].
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