space (X, d), the maps (for each A ⊆ X) d(., A) : (X, d) → (P, d+) are contractions. On P, let e(x, A) = inf{|x − a| : a ∈ A} 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 : U ∈ A}. Let b(U, A) = sup{ inf{ |n − j| : n ∈ X, j ∈ E } : X ∈ U, E ∈ F(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 0 ∈ G, where 0 is the zero metric, that is, 0(x, y) = 0 for all x, y, e ≤ d ∈ G implies e ∈ G, d, e ∈ G implies max(d,e) ∈ G (the "max" here is the pointwise maximum), For all d ∈ XPQ(X), if for all x ∈ X, ε > 0, N < ∞ there is e ∈ G such that min(d(x,y), N) ≤ e(x, y) + ε for all y, then d ∈ G. If G is a gauge on X, then d(x,A) = sup {e(x, a) } : e ∈ G} is a distance function on X. Conversely, given a distance function d on X, the set of e ∈ XPQ(X) such that e ≤ d 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 d ∈ H, d(f(.), f(.)) ∈ G. A tower on X is a set of maps A → A[ε] for A ⊆ X, ε ≥ 0, satisfying for all A, B ⊆ X and δ,