Given a topological space X, can we find another topological space ∇X (ideally in a functorial way…) so that continuous real-valued functions on ∇X coincide with locally constant real-valued functions on X?
mathoverflow.net/q/382011/17064
mathoverflow.net/q/382011/17064
In more sophisticated (i.e., cryptic) terms, does the inclusion functor of the full subcategory of P-spaces (in the sense of Gillman-Henriksen) have a left adjoint?
But with or without sophistication, I don't know whether the ring of locally constant real-valued functions on ℚ can be written as the ring of (all) continuous real-valued functions on some topological space!
Oh, I already received a very nice answer to my question:
‣ the uniform norm can be defined purely algebraically on C(X),
‣ C(X) is always complete for that norm,
‣ but this is not the case for the ring of locally constant functions on ℚ, so it's not a C(X) for any X.
‣ the uniform norm can be defined purely algebraically on C(X),
‣ C(X) is always complete for that norm,
‣ but this is not the case for the ring of locally constant functions on ℚ, so it's not a C(X) for any X.
• • •
Missing some Tweet in this thread? You can try to
force a refresh
