Definition
A subset of Euclidean space is said to be -rectifiable set if there exist a countable collection of continuously differentiable maps
such that the -Hausdorff measure of
is zero. The backslash here denotes the set difference. Equivalently, the may be taken to be Lipschitz continuous without altering the definition.
A set is said to be purely -unrectifiable if for every (continuous, differentiable), one has
A standard example of a purely-1-unrectifiable set in two dimensions is the cross-product of the Smith-Volterra-Cantor set times itself.
Read more about this topic: Rectifiable Set
Famous quotes containing the word definition:
“It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
—Jane Adams (20th century)
“One definition of man is “an intelligence served by organs.””
—Ralph Waldo Emerson (1803–1882)
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (1772–1834)