Subsets of this set named for Vitali do not have a Lebesgue ("luh-BEG") measure, which extends the concept of length to arbitrary subsets of this set.
If U is an open subset of this set, and F is an injective and continuous map from U to this set, then F is a homeomorphism mapping U to the image of F, according to the "invariance of domain" theorem. Subsets of this set named for Vitali do not have a Lebesgue ("luh-BEG") measure, which extends the concept of length to arbitrary subsets of this set. The affinely extended version of this set is compactified by adding points at plus and minus infinity. This set of numbers is the unique ordered field that forms a completion of the (*) rational numbers. There exists no set with a cardinality between the natural numbers and this set according to the continuum hypothesis. This set of numbers is more often formalized using the ZermeloÂ–Fraenkel axioms. Euclidean space is a higher-dimensional version of this set. For 10 points, name this set of numbers that contains both the rationals and irrationals.
If U is an open subset of this set, and F is an injective and continuous map from U to this set, then F is a homeomorphism mapping U to the image of F, according to the "invariance of domain" theorem.
real numbers [accept R; accept Euclidean space or Rn before "Vitali" is read; prompt on "Euclidean space or Rn between "Vitali" and "set of numbers"]