Well, the problem with the union is that if the sets Xi aren’t disjoint then the function f I want above isn’t well-defined. Anyway, as a category theorist at heart I only care about what constructions look like up to natural isomorphism.
Yeah, I should have made more clear, I am aware of that, basically what I was trying to say is I think your paradigm might be inconsistent with what the book is trying to do (not that what the book is trying to do necessarily makes sense). Anyway like I said earlier it is an interesting paradigm, I’d actually never seen it before. By the way regarding the “continuous family of topological spaces” (by which I guess you mean the map sending a point to its fiber under a continuous map), is there a topology on the space of topological spaces such that the family is actually continuous?
Well, the problem with the union is that if the sets Xi aren’t disjoint then the function f I want above isn’t well-defined. Anyway, as a category theorist at heart I only care about what constructions look like up to natural isomorphism.
Yeah, I should have made more clear, I am aware of that, basically what I was trying to say is I think your paradigm might be inconsistent with what the book is trying to do (not that what the book is trying to do necessarily makes sense). Anyway like I said earlier it is an interesting paradigm, I’d actually never seen it before. By the way regarding the “continuous family of topological spaces” (by which I guess you mean the map sending a point to its fiber under a continuous map), is there a topology on the space of topological spaces such that the family is actually continuous?