Fake topological Hilbert spaces and characterizations of dimension in terms of negligibility
Jan J. Dijkstra, Jan van Mill · Fundamenta Mathematicae · 1985
This monograph is an investigation in infinite-dimensional topology.By a fake topological Hilbert space we mean a separable, metrizable space that shares many topological properties with ! 2 , but yet is not homeomorphic to it.We think of properties like: Xis an absolute retract, Xis homogeneous, Xx Xis homeomorphic to l 2 , every compactum in Xis a Z-set and Xis universal for the class of separable, metrizable spaces.Our aim is to construct a sequence X_ 1 ,x 0 ,x 1 , ... of fake Hilbert spaces such that an arbitrary a-compact subspace of~ has dimension~ kif and only if it is strongly negligible.In other words~ has the negligibility properties of l 2 precisely up to dimension k inclusive.The standard way to obtain spaces with certain negligible subsets is through pseudo-boundaries.We first construct in chapter 2 a k-dimensional pseudo-boundary inJR.n.Employing this result we build in chapter 3 a k-dimensional pseudo-boundary in the Hilbert cube for every k E {-1,0,I, ... }.As basis for our sequence x_ 1 ;x 0 ,x 1 , .••we use a fake Hilbert space Y, which has been introduced by Anderson, Curtis & Van Mill [ACM].We show in chapter 4 that Y is homogeneous in a very strong sense and we conclude from this fact that~ is also a pseudo-boundary in Y. Finally, in chapter 5 the spaces Xk = Y\~ are analysed.