Approach space

In topology, approach spaces are a generalization of metric spaces, based on point-to-set distances, instead of point-to-point distances. They were introduced by Robert Lowen in 1989.

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)→[0,∞] 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)→[0,∞] satisfying for all x in X and A, B ⊆ X,

1. d(x,{x}) = 0 ;
2. d(x,{}) = ∞ ;
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 infinum 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[A]) ≤ d(x,A) for all x ∈ X, A ⊆ X.

Lowen has offered at least seven equivalent formulations. Two of them are below.

Let XPQ(X) denote the set of xpq-metrics on X. A subfamily G of XPQ(X) is called a gauge if

1. 0 ∈ G, where 0 is the zero metric, that is, 0(x,y)=0, all x,y ;
2. e ≤ d ∈ G implies e ∈ G ;
3. d, e ∈ G implies max d,e ∈ G (the "max" here is the pointwise maximum);
4. For all d ∈ XPQ(X), if for all x ∈ X, ε>0, N<∞ there is e ∈ G such that min(d(x,y),N) ≤ e(x,y) + ε for all y, then d ∈ G .

If G is a gauge on X, then d(x,A) = sup { e(x,a) } : e ∈ G } is a distance function on X. Conversely, given a distance function d on X, the set of e ∈ XPQ(X) such that e ≤ d is a gauge on X. The two operations are inverse to each other.

A tower on X is a set of maps A→A^[ε] for A⊆X, ε≥0, satisfying for all A, B⊆X, δ, ε ≥ 0

1. A ⊆ A^[ε] ;
2. {}^[ε] = {} ;
3. (A∪B)^[ε] = A^[ε]∪B^[ε] ;
4. A^[ε][δ] ⊆ A^[ε+δ] ;
5. A^[ε] = ∩_δ>εA^[δ] .

Given a distance d, the associated A→A^(ε) is a tower. Conversely, given a tower, the map d(x,A) = inf { ε : x ∈ A^[ε] } is a distance, and these two operations are inverses of each other.

Topological approach spaces

Given a topological space X, one can define a two-valued distance function by d(x,A) = 0 if x ∈ A‾, and ∞ otherwise. Equivalently, the tower is identically the closure operator. One can prove that the only two-valued distance functions come from a topology.

Categorical properties

The main interest in approach spaces and their contractions is that they form a category with good properties, while still being quantitative like metric spaces. One can take arbitrary products and coproducts and quotients, and the results appropriately generalize the corresponding results for topologies. One can even "distancize" such badly non-metrizable spaces like βN, the Stone-Čech compactification of the integers.

Certain hyperspaces, measure spaces, and probabilistic metric spaces turn out to be naturally endowed with a distance. Applications have also been made to approximation theory.

References

R. Lowen Approach Spaces: The Missing Link in the Topology-Uniformity-Metric Triad ISBN 0198500300.