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, d(x, {x}) = 0, d(x, Ø) = ∞, d(x, A∪B) = min(d(x, A), d(x, B)), For all 0 ≤ ε ≤ ∞, d(x, A) ≤ d(x, A(ε)) + ε, where we define A(ε) = {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 → 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 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 ∈ A, 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+(x, A) = max(x − sup A, 0) for x ∈ P and A ⊆ P. Given any approach