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ABSTRACT. The main contribution of this paper is to prove the conjecture of [Va] that the Hilbert cube Q is Lipschitz homogeneous for any metric d3, where s is a decreasing sequence of positive real numbers sjt converging to zero, da(x,y) = sup{sfc|zfc -Vk\-k £ N), and R(s) = sup{sfe/sfc+i: k e N} < oo. In addition to other results, we shall show that for every Lipschitz homogeneous compact metric space X there is a constant A < oo such that X is homogeneous with respect to Lipschitz homeomorphisms whose Lipschitz constants do not exceed A. Finally, we prove that the hyperspace 21 of all nonempty closed subsets of the unit interval is not Lipschitz homogeneous with respect to the Hausdorff metric.

Abstract.We show that a uniform space iiA" is supercomplete if, and only if, the Ginsburg-Isbell locally fine coreflection of p.X X 'Sß X is equinormal.1. Introduction. J. Isbell calls a uniform space pX supercomplete if the uniform hyperspace H(pX) of closed subsets of X is complete. He proved in [4] that pX is supercomplete if, and only if, X is paracompact and Xp = ®sx, where ^Fx is the fine uniformity of X. (On the other hand, Morita proved in [7] that the uniform hyperspace K(p.X) of all compact subsets of X is complete if and only if p. X is complete.) Tamaño established in 1960 (see [9]) the following famous result: a completely regular space X is paracompact whenever X X ßX is normal. It seems reasonable to ask if it is possible to give a Tamano-like "finitary" characterization of supercomplete uniform spaces by means of their Cech-Stone compactifications. First of all, we will recall some familiar definitions. A quasiuniformity is a filter of coverings ordered by the relation of refinement. If p and v are quasiuniformities, then p/v is the quasiuniformity consisting of all covers {V, D IV'}, where {V¡}¡ G v and for each i, [Wf}j G p. If p is a quasiuniformity, let ju(0) = p. If //0) is defined for all a < ß and ß is a limit ordinal, let p{ß) = U {pia): a < ß). On the other hand, if ß -a + 1, then we define p(ß) = p{a)/p

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