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}. 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,

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.

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.