Approach space

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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 ∞pq-metric here), one can define an induced map d:X×P(X)→[0,∞] by d(x,A) = inf { d(x,a ) : aA }. 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, BX,

1. d(x,{x}) = 0 ;
2. d(x,Ø) = ∞ ;
3. d(x,AB) = 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[A]) ≤ d(x,A) for all xX, AX.

Examples

Every ∞pq-metric space (X,d) can be distancized to (X,d), as described at the beginning of the definition.

Given a set X, the discrete distance is given by d(x,A) = 0 if xA and = ∞ if xA. The induced topology is the discrete topology.

Given a set X, the indiscrete distance is given by d(x,A) = 0 if A is non-empty, and = ∞ if A is empty. The induced topology is the indiscrete topology.

Given a topological space X, a topological distance is given by d(x,A) = 0 if xA, and = ∞ if not. The induced topology is the original topology. In fact, the only two-valued distances are the topological distances.

Let P=[0,∞], the extended positive reals. Let d+(x,A) = max (x−sup A,0) for xP and AP. Given any approach space (X,d), the maps (for each AX) d(.,A) : (X,d) → (P,d+) are contractions.

On P, let e(x,A) = inf { |xa| : aA } for x<∞, let e(∞,A) = 0 if A is unbounded, and let e(∞,A) = ∞ if A is bounded. Then (P,e) is an approach space. Topologically, P is the one-point compactification of [0,∞). Note that e extends the ordinary Euclidean distance. This cannot be done with the ordinary Euclidean metric.

Let βN be the Stone–Čech compactification of the integers. A point U∈βN is an ultrafilter on N. A subset A⊆βN induces a filter F(A)=∩{U:UA}. Let b(U,A) = sup { inf { |n-j| : nX, jE } : XU, EF(A) }. Then (βN,b) is an approach space that extends the ordinary Euclidean distance on N. In contrast, βN is not metrizable.

Equivalent definitions

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. edG implies eG ;
3. d, eG implies max d,eG (the "max" here is the pointwise maximum);
4. For all d ∈ XPQ(X), if for all xX, ε>0, N<∞ there is eG such that min(d(x,y),N) ≤ e(x,y) + ε for all y, then dG .

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

A contraction f:(X,d)→(Y,e) is, in terms of associated gauges G and H respectively, a map such that for all dH, d(f(.),f(.))∈G.

A tower on X is a set of maps AA[ε] for AX, ε≥0, satisfying for all A, BX, δ, ε ≥ 0

1. AA[ε] ;
2. Ø[ε] = Ø ;
3. (AB)[ε] = A[ε]B[ε] ;
4. A[ε][δ]A[ε+δ] ;
5. A[ε] = ∩δ>εA[δ] .

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

A contraction f:(X,d)→(Y,e) is, in terms of associated towers, a map such that for all ε≥0, f[A[ε]] ⊆ f[A][ε].

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

• Lowen, Robert (1997). Approach spaces: the missing link in the topology-uniformity-metric triad. Oxford Mathematical Monographs. Oxford: Clarendon Press. ISBN 0-19-850030-0. Zbl 0891.54001.
• Lowen, Robert (2015). Index Analysis: Approach Theory at Work. Springer.