A factorization of the direct limit of Hilbert cubes

Abstract:

We show that the countable direct limit of Hilbert cubes ${Q^\infty }$ is homeomorphic to the product of the Hilbert cube $Q$ and the countable direct limit of lines ${R^\infty }$. As a consequence, two open subsets of ${R^\infty }$ 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 $X \times Q \times {R^\infty } \cong Q \times {R^\infty }$, where $X$ is any locally compact, separable AR (metric).