Intopology, a branch ofmathematics,approach spaces are a generalization ofmetric spaces, based on point-to-set distances, instead of point-to-point distances. They were introduced by Robert Lowen in 1989, in a series of papers on approach theory between 1988 and 1995.
Given a metric space (X,d), or more generally, anextendedpseudoquasimetric (which will be abbreviated∞pq-metric here), one can define an induced mapd:X × P(X) → [0,∞] byd(x,A) =inf{d(x,a) :a ∈A}. With this example in mind, adistance onX is defined to be a mapX × P(X) → [0,∞] satisfying for allx inX andA,B ⊆X,
where we defineA(ε) = {x :d(x,A) ≤ ε}.
(The "empty infimum is positive infinity" convention is like thenullary intersection is everything convention.)
An approach space is defined to be a pair (X,d) whered is a distance function onX. Every approach space has atopology, given by treatingA → A(0) as aKuratowski closure operator.
The appropriate maps between approach spaces are thecontractions. A mapf: (X,d) → (Y,e) is a contraction ife(f(x),f[A]) ≤d(x,A) for allx ∈X andA ⊆X.
Every ∞pq-metric space (X,d) can bedistanced to (X,d), as described at the beginning of the definition.
Given a setX, thediscrete distance is given byd(x,A) = 0 ifx ∈A andd(x,A) = ∞ ifx ∉A. Theinduced topology is thediscrete topology.
Given a setX, theindiscrete distance is given byd(x,A) = 0 ifA is non-empty, andd(x,A) = ∞ ifA is empty. The induced topology is the indiscrete topology.
Given atopological spaceX, atopological distance is given byd(x,A) = 0 ifx ∈A, andd(x,A) = ∞ otherwise. The induced topology is the original topology. In fact, the only two-valued distances are the topological distances.
LetP = [0, ∞] be theextended non-negativereals. Letd+(x,A) = max(x −supA, 0) forx ∈P andA ⊆P. Given any approach space (X,d), the maps (for eachA ⊆X)d(.,A) : (X,d) → (P,d+) are contractions.
OnP, lete(x,A) = inf{|x −a| :a ∈A} forx < ∞, lete(∞,A) = 0 ifA is unbounded, and lete(∞,A) = ∞ ifA is bounded. Then (P,e) is an approach space. Topologically,P is the one-point compactification of [0, ∞). Note thate extends the ordinary Euclidean distance. This cannot be done with the ordinary Euclidean metric.
Let βN be the Stone–Čech compactification of theintegers. A pointU ∈ βN is an ultrafilter onN. A subsetA ⊆ βN induces a filterF(A) = ∩ {U :U ∈A}. Letb(U,A) = sup{ inf{ |n −j| :n ∈X,j ∈E } :X ∈U,E ∈F(A) }. Then (βN,b) is an approach space that extends the ordinary Euclidean distance onN. In contrast, βN is not metrizable.
Lowen has offered at least seven equivalent formulations. Two of them are below.
Let XPQ(X) denote the set of xpq-metrics onX. A subfamilyG of XPQ(X) is called agauge if
IfG is a gauge onX, thend(x,A) = sup {e(x,a) } :e ∈G} is a distance function onX. Conversely, given a distance functiond onX, the set ofe ∈ XPQ(X) such thate ≤d is a gauge onX. The two operations are inverse to each other.
A contractionf: (X,d) → (Y,e) is, in terms of associated gaugesG andH respectively, a map such that for alld ∈H,d(f(.),f(.)) ∈G.
Atower onX is a set of mapsA →A[ε] forA ⊆X, ε ≥ 0, satisfying for allA,B ⊆X and δ, ε ≥ 0
Given a distanced, the associatedA →A(ε) is a tower. Conversely, given a tower, the mapd(x,A) = inf{ε :x ∈A[ε]} is a distance, and these two operations are inverses of each other.
A contractionf:(X,d)→(Y,e) is, in terms of associated towers, a map such that for all ε ≥ 0,f[A[ε]] ⊆f[A][ε].
The main interest in approach spaces and their contractions is that they form acategory with good properties, while still being quantitative like metric spaces. One can take arbitraryproducts,coproducts, and quotients, and the results appropriately generalize the corresponding results for topologies. One can even "distancize" such badly non-metrizable spaces like βN, theStone–Čech compactification of the integers.
Certain hyperspaces,measure spaces, andprobabilistic metric spaces turn out to be naturally endowed with a distance. Applications have also been made toapproximation theory.