A Q-INFINITY-MANIFOLD TOPOLOGY OF THE SPACE OF LIPSCHITZ-MAPS

Let X be a nondiscrete metric compactum and Y an Euclidean polyhedron without isolated Points or a Lipschitz n-manifold (n > 0). Let LIP(X, Y)w be the space of Lipschitz maps from X to Y admitting the weak topology with respect to the tower 1-LIP(X, Y) subset-of 2-LIP(X, Y) subset-of 3-LIP(X, Y) subset-of ..., where k-LIP(X, Y) denotes the space of Lipschitz maps with Lipschitz constant less-than-or-equal-to k with the sup-metric. It is proved that LIP(X, Y)w is a manifold modeled on Q(infinity) = dir lim Q(n), the direct limit of Hilbert cubes.