Approach Space - Equivalent Definitions

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

1. 0 ∈ G, where 0 is the zero metric, that is, 0(x,y)=0, all x,y ;
2. e ≤ d ∈ G implies e ∈ G ;
3. d, e ∈ G implies max d,e ∈ G (the "max" here is the pointwise maximum);
4. 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 for A⊆X, ε≥0, satisfying for all A, B⊆X, δ, ε ≥ 0

1. A ⊆ A ;
2. Ø = Ø ;
3. (A∪B) = A∪B ;
4. A ⊆ A ;
5. A = ∩_δ>εA .

Given a distance d, the associated A→A(ε) is a tower. Conversely, given a tower, the map d(x,A) = inf { ε : x ∈ A } is a distance, and these two operations are inverses of each other.

A contraction f:(X,d)→(Y,e) is, in terms of associated towers, a map such that for all ε≥0, f] ⊆ f.