By Oscar Zariski

Zariski offers an effective advent to this subject in algebra, including his personal insights.

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Note also that x | drn − drn | < ε < 1/2. As a consequence, one cannot strongly amalgamate two copies of Yn ∪ {x} by gluing the two copies of Yn together while working with distances in ω only. 3. URYSOHN SPACES 27 Indeed, assume that Yn ∪ {x, x } is such an amalgam. Then x−x x + x < 1. The same argument also exhibits a negative amalgamation property for most of the classes SS when S = {k/m : k ∈ {1, . . , 2m}}. Namely, it shows that that there is M ∈ ω such that for every integer m M , the class SS does not have the strong amalgamation property.

Then, set: s0 = dY0 (y0 , y), s1 = dY1 (y1 , y) s0 = dY0 (y0 , y ), s1 = dY1 (y1 , y ) Set also: t = dY0 (y, y ) = dY1 (y, y ). Then observe that: |s0 − s1 | t s0 + s1 , |s0 − s1 | t s0 + s1 . 22 ´ CLASSES OF FINITE METRIC SPACES AND URYSOHN SPACES 1. FRA¨ISSE So by the 4-values condition, we obtain the required u ∈ S. We now proceed by induction on the size of the symmetric diﬀerence Y0 ΔY1 . The previous proof covers the case |Y0 ΔY1 | 2. For the induction step, let Y = Y0 ∪ Y1 . The cases where Y0 and Y1 are ⊂-comparable are obvious, so we may assume that Y0 and Y1 are ⊂-incomparable.

Seeing Yn as a subset of Rn−1 with isobarycentre 0 2 , let x ∈ Rn be orthogonal to Rn−1 and such that: ∀y ∈ Yn x − y = drn . Then Yn ∪ {x} ∈ Hω . Note also that x | drn − drn | < ε < 1/2. As a consequence, one cannot strongly amalgamate two copies of Yn ∪ {x} by gluing the two copies of Yn together while working with distances in ω only. 3. URYSOHN SPACES 27 Indeed, assume that Yn ∪ {x, x } is such an amalgam. Then x−x x + x < 1. The same argument also exhibits a negative amalgamation property for most of the classes SS when S = {k/m : k ∈ {1, .