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.
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}.
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.
Let d+(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) : (X, d) → (P, d+) are contractions.
One can even "distancize" such badly non-metrizable spaces like βN, the Stone–Čech compactification of the integers.