In two interesting posts(here and here),it is mentioned that "there is a close connection between semi-decidable logics and topological spaces"
Michael O’Connor wrote:
In fact, given a topological space, you can form a semi-decidable logic by making a propositional symbol $P_U$ for each open set $U$ . You can then interpret all propositional formulas as open sets by interpreting $\lor$ as union and $\land$as intersection. Finally, take as a set of axioms the set of all statements $P \vdash Q$where the open set corresponding to $P$is a subset of the open set corresponding to $P$ . This set is closed under the inference rules given.
You can also start with a semi-decidable logic and generate a topology; this is a form of Stone duality. In general, if you start with a topological space, translate to a semi-decidable logic, then translate back, you might not get your original space back. However, you will if the space you start with is sufficiently nice (e.g., Hausdorff).
Question: Could anyone give a concrete example of how this can be done? For example what's semidecidable logic of $\omega^{\omega}$ with a product topology? What is $P_U$, if $U = \{0\} \times \omega^{\omega}$?
Any reference on this topic is also welcome. In a comment,Peter Berry suggested "the book “Topology via Logic” by Stephen Vickers takes this approach". So I think this book must be a good reference on this problem, but I can't find the keyword semi-decidable in its index. This book can be found here on googlebooks.
