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. edG implies eG ;
  3. d, eG implies max d,eG (the "max" here is the pointwise maximum);
  4. For all d ∈ XPQ(X), if for all xX, ε>0, N<∞ there is eG such that min(d(x,y),N) ≤ e(x,y) + ε for all y, then dG .

If G is a gauge on X, then d(x,A) = sup { e(x,a) } : eG } is a distance function on X. Conversely, given a distance function d on X, the set of e ∈ XPQ(X) such that ed 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 dH, d(f(.),f(.))∈G.

A tower on X is a set of maps AA for AX, ε≥0, satisfying for all A, BX, δ, ε ≥ 0

  1. AA ;
  2. Ø = Ø ;
  3. (AB) = AB ;
  4. AA ;
  5. A = ∩δ>εA .

Given a distance d, the associated AA(ε) is a tower. Conversely, given a tower, the map d(x,A) = inf { ε : xA } 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.

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