Approach Space - Definition

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,

1. d(x,{x}) = 0 ;
2. d(x,Ø) = ∞ ;
3. d(x,A∪B) = min d(x,A),d(x,B) ;
4. 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.