A factorization of the direct limit of Hilbert cubes

Richard E. Heisey
1976 Proceedings of the American Mathematical Society  
We show that the countable direct limit of Hilbert cubes 0°° is homeomorphic to the product of the Hilbert cube Q and the countable direct limit of lines R°°. As a consequence, two open subsets of Rx have the same homotopy type if and only if their products with Q are homeomorphic. Combined with a theorem of D. W. Henderson our result implies that XxQxR°°sQx Rx, where X is any locally compact, separable AR (metric). At the problem session at the Conference on Infinite-Dimensional Topology at
more » ... onal Topology at Louisiana State University, October 1973, Tom Chapman asked if Rx X Q =s <2°°. Here 7?°° = dir lim R", Qx = dir lim Q", where 7? denotes the reals and Q denotes the Hilbert cube, which we regard as the countably infinite product of 7 = [0,1], (We remark that if B is any separable, infinitedimensional Banach space, then its conjugate B* with the bounded weak-* topology is homeomorphic to (9°° [4,.) We answer the above question in the affirmative. As a corollary we show (Corollary 5) that two open subsets of 7?°° have the same homotopy type iff their products with Q are homeomorphic. Finally, combining our result with work of D. W. Henderson [6] we obtain a factorization result for locally compact, separable AR's and
doi:10.1090/s0002-9939-1976-0418102-9 fatcat:eooofhzbvrfq5bgdeouxgud4aa