Suppose I have a diagram that consists of a small circle, call it W, within a larger circle, call it G. (Note: this a trivial doughnut chart, which is a kind of pie chart, so the circles aren’t literally circles.)
Further suppose that:
(i) G is a non-metric space (i.e. notions of distance or area don’t apply). (ii) W is a metric space (iii) G “overlaps” W, in the sense that if W were removed, there is no hole: all of G would still be there (cf. the analogy in the addendum below)
I think that answers given to another question have already established that the conjunction of (i) & (ii) is consistent.
What I want to know is whether the conjunction of (i), (ii), & (iii) is consistent.
In particular, I want to know if there is a topological (or mereotopological) way of representing the situation so that G, despite its non-metrical nature, can coexist and interface with W in virtue of its overlapping W in the sense of (iii).
Furthermore, I want to be able to say that where G "overlaps" with W in the specified sense, it can be said to exist in (throughout) W, without that fact being affected by the metric nature of W. That is to say, I don't want my conditions to entail that there is more (or less) of G in different-sized regions (subspaces) of W. Is that possible?
Addendum: Here’s a physical analogy that may help clarify what I’m trying to express. Instead of circle G imagine a large, thin, and transparent sheet of yellow film, and instead of circle W, imagine a thin transparent blue disk-shaped piece of film. If I place the blue film under or over the yellow film (so that there is an interface between blue and yellow areas) I will have an area that is green, although the corresponding blue and yellow areas in a sense also both still exist (i.e. coexist) as well. And if I remove the blue disk, the yellow area under or over it reappears (it was never really gone).
If we let yellow represent G’s property of being topologically non-metric and blue represent W’s property of being topologically metric, is there a way of saying that the topological properties can coexist as blue and yellow do, and that there is also way of interfacing (like the one that produces green) that may merge both topological properties rather than merely eliminate one or the other?