Definition
Given a metric space (X,d), or more generally, an extended pseudoquasimetric (which will be abbreviated xpq-metric here), one can define an induced map d:X×P(X)→ by d(x,A) = inf { d(x,a ) : a ∈ A }. With this example in mind, a distance on X is defined to be a map X×P(X)→ satisfying for all x in X and A, B ⊆ X,
- d(x,{x}) = 0 ;
- d(x,Ø) = ∞ ;
- d(x,A∪B) = min d(x,A),d(x,B) ;
- For all ε, 0≤ε≤∞, d(x,A) ≤ d(x,A(ε)) + ε ;
where A(ε) = { x : d(x,A) ≤ ε } by definition.
(The "empty infimum is positive infinity" convention is like the nullary intersection is everything convention.)
An approach space is defined to be a pair (X,d) where d is a distance function on X. Every approach space has a topology, given by treating A → A(0) as a Kuratowski closure operator.
The appropriate maps between approach spaces are the contractions. A map f:(X,d)→(Y,e) is a contraction if e(f(x),f) ≤ d(x,A) for all x ∈ X, A ⊆ X.
Read more about this topic: Approach Space
Famous quotes containing the word definition:
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animalsjust as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)
“Its 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)