Definable functions in Urysohn’s metric space
نویسندگان
چکیده
منابع مشابه
Definable Functions in Urysohn’s Metric Space
Let U denote the Urysohn sphere and consider U as a metric structure in the empty continuous signature. We prove that every definable function U → U is either a projection function or else has relatively compact range. As a consequence, we prove that many functions natural to the study of the Urysohn sphere are not definable. We end with further topological information on the range of the defin...
متن کاملAn approximate Herbrand's theorem and definable functions in metric structures
We develop a version of Herbrand’s theorem for continuous logic and use it to prove that definable functions in infinite-dimensional Hilbert spaces are piecewise approximable by affine functions. We obtain similar results for definable functions in Hilbert spaces expanded by a group of generic unitary operators and Hilbert spaces expanded by a generic subspace. We also show how Herbrand’s theor...
متن کاملReal Linear-Metric Space and Isometric Functions
Let V be a non empty metric structure. We say that V is convex if and only if the condition (Def. 1) is satisfied. (Def. 1) Let x, y be elements of the carrier of V and r be a real number. Suppose 0 ¬ r and r ¬ 1. Then there exists an element z of the carrier of V such that ρ(x, z) = r · ρ(x, y) and ρ(z, y) = (1 − r) · ρ(x, y). Let V be a non empty metric structure. We say that V is internal if...
متن کاملInteger-valued definable functions
We present a dichotomy, in terms of growth at infinity, of analytic functions definable in the real exponential field which take integer values at natural number inputs. Using a result concerning the density of rational points on curves definable in this structure, we show that if a function f : [0,∞) → R is such that f(N) ⊆ Z, then either sup|x̄|≤r f(x̄) grows faster than exp(r), for some δ > 0,...
متن کاملDefinable Smoothing of Continuous Functions
Let R be an o-minimal expansion of a real closed field. Given definable continuous functions f : U → R and : U → (0,+∞), where U is an open subset of Rn, we construct a definable Cm-function g : U → R with |g(x)− f(x)| < (x) for all x ∈ U . Moreover, we show that if f is uniformly continuous, then g can also chosen to be uniformly continuous.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Illinois Journal of Mathematics
سال: 2011
ISSN: 0019-2082
DOI: 10.1215/ijm/1373636691