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Dear stackexchangicians,

I have been reading an expository paper about the information theory founded by C. Shannon. The following question is vague, but has been there successful applications of information theory to study of sets and topological spaces? Given a “nice” topological space like compact, connected space, is there a way to study “information” for open sets and their flow among the network of open sets? For example, can we describe an information about the open set and if that can be transferable to another open set or even some subsets of the same topological space?

Andrés E. Caicedo
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Phoenix13
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    IMHO, a general topological structure, even compact/connected is too poor. You need a richer underlying structure (at least semi-norms on a topological vector space) in order to be able to "measure" something. – Jean Marie May 10 '20 at 22:33
  • @Jean Marie. Thanks! The topological space I have been investigating is built from the set of all integers, which I think do not admit those useful structures. Can we define useful norm or metric to integers? – Phoenix13 May 10 '20 at 22:36
  • Yes p-adic metrics. See for example my recent answer here. – Jean Marie May 10 '20 at 22:39
  • I am not sure if it's what you're looking for, but have you heard of descriptive set theory? – Jonathan Schilhan May 11 '20 at 10:16
  • If you allow "information theory" as being broadly enough formulated to encompass ergodic theory, then there are many many connections between ergodic theory and topology. In ergodic theory, one studies a continuous self-map $f : X \to X$ of a topological space by focussing on the set of $f$-invariant Borel probability measures, and one analyzes those measures using ideas of information theory, for example using partitions of $X$ into open sets, and the networks amongst those open sets that are induced by the map $f$. – Lee Mosher May 11 '20 at 13:12
  • @Lee Mosher That is interesting to learn! The only problem is that the topology I have in mind is built on integers, which I am not sure if measure theory could be applied. – Phoenix13 May 11 '20 at 13:45

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