Still, I wonder if there exist space-filling surfaces that are limits of surfaces S_n in R^3 as n —> oo, such that ... if for some eps > 0, the epsilon neighborhood of S_n (meaning {x in R^3 | ||x-p|| < eps for some p in S_n}) is denoted by N_eps(S_n), then *hypothetically*: N_eps(S_n) contains N_(1/eps)(0) which we require must hold for all n, for some sequence of eps = eps_n > 0 that decreases to 0 as n —> oo. And of course we require that for some sequence h_n of embeddings h_n : R^2 —> R^3 of the plane into R^3 with h_n(R^2) = S_n, we have that the limit lim_{n—>oo} h_n exists and is continuous. It's space-filling by the hypothetical construction. —Dan Andy Latto wrote ----- Aren't there even simpler solutions? If f:R -> R^3 is space-filling, define g:R^2 -> R^3 by g(x,y) = f(x).
On Thu, Dec 13, 2018 at 4:34 PM Bill Gosper <billgosper@gmail.com> wrote:
Rohan requested a continuous map from a 2D surface onto a non-porous patch of positive 3D volume.