Alexandrov topology

The Alexandrov topology is a topology that can be put on any poset.

Definition

Let (P, \sqsubseteq) be a poset. Then the open sets of the Alexandrov topology are the upper sets of P, i.e. the subsets S \sqsubseteq P such that

  p \in S\ \land\ p \sqsubseteq q
  \Longrightarrow\
  q \in S

Intuition

Think of a poset (P, \sqsubseteq) as consisting of some "finite pieces of information" and of p \sqsubseteq q as meaning that q has at least as much information as p.

Then, each open set S of the Alexandrov topology can be seen as one that is closed under information extension: if p \in S then S also contains all elements that have more information than p. If we can observe p in a finitistic way (e.g. in finite time), then we can establish the property of being in S in a finite way.

Thus, the Alexandrov topology can be seen as capturing a logic of finite observation.

If the elements of P are to be seen as infinitary observations, then the Scott topology is more appropriate instead.

These intuitions come from Vickers 1989, Chapter 3.

References

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