The analogy goes much further if you take it as "open set" =~ "semidecidable property", i.e., you can determine in finite time that something is in the subset, but may not halt on an input that is not in the subset. A closed set is one whose complement is semidecidable, i.e., "co-semidecidable". This gives some intuition to the finite intersection vs arbitrary union property (though in computability you would restrict to a kind of "effective" union, but still one that could be infinite). Then sets that are closed and open are the decidable properties.
Then the inverse-image definition of a continuous function is clearly satisfied by computable functions. A function f : A -> B is continuous if for any semidecidable property O on B, the property { x \in A. f(x) \in O } is semidecidable (since f is a computable function).
There's some very cool ideas here like Hausdorff == Diagonal is closed =~ the equality predicate is co-semidecidable, i.e., you can implement a partial inequality function that always returns true if the inputs are not equal.
Then the inverse-image definition of a continuous function is clearly satisfied by computable functions. A function f : A -> B is continuous if for any semidecidable property O on B, the property { x \in A. f(x) \in O } is semidecidable (since f is a computable function).
There's some very cool ideas here like Hausdorff == Diagonal is closed =~ the equality predicate is co-semidecidable, i.e., you can implement a partial inequality function that always returns true if the inputs are not equal.
See Martín Escardó's work for more details: https://www.cs.bham.ac.uk/~mhe/papers/barbados.pdf