Open Subsets of LF-spaces
Volume 56 / 2008
Bulletin Polish Acad. Sci. Math. 56 (2008), 25-37
MSC: 46A13, 46T05, 57N17, 57N20.
DOI: 10.4064/ba56-1-4
Abstract
Let $F = \mathop{\rm ind\,lim} F_n$ be an infinite-dimensional LF-space with density $\mathop{\rm dens} F = \tau$ ($\geq \aleph_0$) such that some $F_n$ is infinite-dimensional and $\mathop{\rm dens} F_n = \tau$. It is proved that every open subset of $F$ is homeomorphic to the product of an $\ell_2(\tau)$-manifold and $\mathbb R^\infty = \mathop{\rm ind\,lim} \mathbb R^n$ (hence the product of an open subset of $\ell_2(\tau)$ and $\mathbb R^\infty$). As a consequence, any two open sets in $F$ are homeomorphic if they have the same homotopy type.
