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Approach space

In topology, a branch of mathematics, 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, in a series of papers on approach theory between 1988 and 1995.

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) : 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, Ø) = ∞,
  3. d(x, A∪B) = min(d(x, A), d(x, B)),
  4. For all 0 ≤ ε ≤ ∞, d(x, A) ≤ d(x, A<sup>(ε)</sup>) + ε,

where we define A<sup>(ε)</sup> = {x : d(x, A) ≤ ε}.

(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&nbsp;→&nbsp;A<sup>(0)</sup> 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 and A ⊆ X.

Examples

Every ∞pq-metric space (X, d) can be distanced 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 x ∈ A and d(x, A) = ∞ if x ∉ A. 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 d(x, A) = ∞ 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 x ∈ <span style="text-decoration: overline;">A</span>, and d(x, A) = ∞ otherwise. The induced topology is the original topology. In fact, the only two-valued distances are the topological distances.

Let P = [0, ∞] be the extended non-negative reals. Let d<sup>+</sup>(x, A) = max(x − sup A, 0) for x ∈ P and A ⊆ P. Given any approach space (X, d), the maps (for each A ⊆ X) d(., A)&nbsp;:&nbsp;(X, d)&nbsp;→&nbsp;(P, d<sup>+</sup>) are contractions.

On P, let e(x, A) = inf